I 


00 


en 


^ 


M 


im 


ft* 


rt 


<i    c-.. 


V-. 


IN  MEMORIAM 
FLORIAN  CAJORl 


^K>^ 


V 


,  ~i 


PREFACE. 


n->n'h'rn^lci 


However  forcibly  an  author  may  be  impressed  with 
the  conviction  that  his  work  constitutes  an  important 
improvement  on  all  similar  efforts  which  have  preceded 
it,  he  is  still  aware  that  its  favorable  reception  by  the 
public  depends  upon  the  recognition  of  its  merits  by 
other  minds  than  his  own. 

In  regard  to  this  book,  if  the  improvements  which 
I  have  attempted  to  incorporate  into  it  are  not  readily 
recognized  by  the  experienced  teacher  as  he  peruses  it, 
they  are  certainly  of  so  little  value  as  not  to  be  worth 
pointing  out.  I  have,  then,  only  to  suggest  to  the 
reader  to  turn  to  the  subjects,  say  of  Ratio  and  Pro- 
portion, and  critically  peruse  the  articles  as  they  occur, 
including  the  examples  in  the  application  of  principles 
and  the  references.  If  I  mistake  not,  a  few  pages  will 
reveal  to  him  many  of  the  iinportant  features  which 
distinguish  this  work;  and,  if  an  experience  of  nea.rly 
thirty  years  in  the  school-room  justifies  me  in  express- 
ing the  opinion,  he  will  be  surprised  that  they  have  not 
been  hitherto  developed. 


q-v 


PREFACE. 


There  is  only  one  point  to  which  I  will  expressly 
refer.  It  will  be  seen  that  decimal  fractions  are  the 
orflfepring  of  decimal  notation,  and  not  of  ^^vidgar" 
fractions,  and  that  their  notation  is  early  introduced  for 
the  sake  of  scientific  accuracy,  as  well  as  the  early  in- 
sertion of  problems  involving  United  States  currency; 
for  if  there  is  any  concrete  quantity  which  the  Amer- 
ican   child    readily   understands,   it   is    that   involving 

dollars  and  cents. 

P.  A.  TOWNE. 
Mobile,  Ala.,  January^  1866. 


0  r^h), 


CONTENTS 


a 


PAGE 

Definitions 9 

Notation 11 

Arabic  Notation 11 

Numeration 17 

Decimals 20 

Notation  and  Numeration 20 

Principles  of  Arabic  Notation 25 

United  States  Money — Notation  and  Numeration 26 

Roman  Notation  and  Numeration 30 

Addition 31 

Subtraction 45 

Multiplication 55 

Division 69 

Short  Division 71 

Long  Division 78 

Properties  of  Integral  Numbers 87 

Definitions 87 

Factoring 91 

Greatest  Common  Divisor 93 

Least  Common  Multiple 99 

Fractions 108 

Nature  of  Fractions 108 

Notation  of  Fractions 109 

Classification  of  Fractions 112 

Value  of  a  Fraction 113 

Propositions  in  Fractions 113 

Pteduction  of  Fractions 113 

(•5) 


% 


♦ 


^■yt  t-ti'^ 


6  '   "    *■-**        ■  COSTHNIS."" 


D 

Fractions — [Continued.)  pagk 

Addition  of  Fractions 119 

Subtraction  of  Fractions 121 

Multiplication  of  Fractions 124 

Division  of  Fractions 130 

Reduction  of  Common  Fractions  to  Decimal  Fractions 136 

Compound  Numbers 151 

Definitions 151 

English  Money 152 

French  Money ;  152 

Troy  Weight 153 

Avoirdupois  Weight 154 

Apothecaries  Weight 155 

Long  Measure 155 

Cloth  Measure 156 

Superficial  or  Square  Measure 158 

Solid  Measure 159 

Wine  Measure 160 

Ale  or  Beer  Measure 161 

Dry  Measure 161 

Time 162 

Circular  Measure 164 

Reduction  of  Compound  Numbers 166 

Compound  to  Concrete 166 

Concrete  to  Compound 166 

Denominate  Fractions  to  Compound  Numbers 169 

Compound  Numbers  to  Denominate  Fractions 169 

Compound  Numbers  to  Decimal  Fractions 173 

Denominate  Decimal  Fractions  to  Compound  Numbers 173 

Addition  of  Compound  Numbers 175 

Subtraction  of  Compound  Numbers 177 

Time  between  Dates 179 

Multiplication  of  Compound  Numbers 181 

Division  of  Compound  Numbers 184 

Longitude  in  Time 186 

Analysis  by  Aliquot  parts 187 

Review  in  Addition 195 

Review  in  Subtraction 198 

Review  in  IMultiplicatiou  and  I'ivision 200 


CONTENTS.  -^  7 


PAGE 

Percentage 202 

Applications  of  Percentage 210 

Insurance 210 

Commission 212 

Stock 213 

Brokerage 215 

Profit  and  Loss 215 

Duties  or  Customs 220 

Interest 222 

Problems  in  Interest 232 

Present  Worth *....  235 

Bank  Discount 236 

Promissory  Notes 237 

Compound  Interest 241 

Ratio 244 

Proportion 246 

Rule  of  Three 250 

Partnership 261 

Equation  of  Payments 265 

Alligation  Medial 266 

Alligation  Alternate 268 

Position 273 

Single  Position 273 

Double  Position 276 

Involution 281 

Evolution 285 

Square  Root 286 

Cube  Root 295 

Problems 303 

Arithmetical  Progression 305 

Geometrical  Progression 311 

Permutations,  Arrangements,  and  Combinations 317 

Practical  Geometry 319 

Definitions , 319 

Pythagorean  Proposition 323 

Proposition  on  the  Triangle 326 

Mensuration 327 

Area  of  Triangle,  I...^ 327 


8  CONTENTS. 

a 

Practical  Geometry — [Continued.)  page 

Area  of  Triangle,  II 328 

Area  of  Quadrilateral  with  Parallel  Sides 328 

Area  of  Trapezium 329 

Proposition  on  Similar  Figures 330 

The  Grindstone  Problem 331 

Mensuration  of  Solids 332 

Definition 332 

Contents  of  a  Cylinder  or  Prism 333 

Contents  of  a  Cone  or  Pyramid 334 

Contents  of  a  Frustrum  of  a  Cone  or  Pyramid 335 

ConCents  of  a  Cistern 335 

Proposition  on  Spheres 335 

Surface  and  Contents  of  a  Sphere 336 

Miscellaneous  Examples 336 

Appendix 351 

Table  of  Multiplication 351 

Table  of  Square  Roots 362 

Table  of  Cube  Roots 353 

Strength  of  Building  Materials 354 

Annuities  •. 355 

French  Weight 357 

French  Linear  Measure 357 

French  Superficial  Measure 358 

French  Solid  Measure 358 

French  Measure  of  Capacity 358 

Table  of  Foreign  Money  (fixed  by  law) 358 

Books  and  Paper 360 


ELEMENTARY   ARITHMETIC. 


DEFINITIONS 


1.  Science  is  knowledge  reduced  to  order. 

2.  Art  is  the  practical  application  of  the  principles 
of  a  science. 

3.  Quantity  is  a  term  that  is  applied  to  any  thing 
that  can  be  measured. 

4.  A  Unit  is  a  quantity  to  which  the  term  one  may 
be  applied.  Thus:  one  horse,  one  ten,  one  half. — 
(Vide  128.) 

5.  A  Number  is  a  unit,  or  a  collection  of  units. 

6.  Figures  are  characters  used  to  represent  any  given 
number.  They  include  the  cipher,  naught,  or  zero,  and 
nine  digits.     Thus : 


naught 

one 

two 

three 

four 

five 

six 

seven 

eight 

nine 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

The  ni7ie  digits  are  called  significant  figures,  to  distin- 
guish them  from  the  cipher,  which  has,  when  written 
alone,  no  value  whatever.  Its  effect  when  joined  to 
other  figures  is  explained  under  Notation. 

7.  Integral  numbers  or  integers  are  tvhole  numbers. 
Thus:   seven,  twenty,  one  hundred  and  six,  etc.,  are 


integral  numbers. 


(i>) 


10  DEFINITIONS. 

8.  A  Fractional  number  or  fraction  represents  one, 
or  more  than  one,  of  the  equal  parts  of  a  unit.  Thus : 
one  seventh,  five  twentieths,  two  thirds,  four  ninths,  etc., 
<ire  fractional  numbers  or  fractions. — (Vide  127.) 

O.  A  Decimal  fraction  represents  07ie,  or  more  than 
one,  of  the  parts  of  a  unit  which  is  divided  into  ten,  one 
hundred,  one  thousand,  ten  thousand,  etc.,  equal  parts. 
Thus :  one  tenth,  four  tenths,  thirty-seven  hundredths, 
forty-five  thousandths,  etc.,  are  decimal  fractions. 

10.  Mathematics  is  the  science  of  quaiitity. 

11.  Arithmetic  is  that  branch  of  mathematics  which 
treats  of  numbers.     It  is  a  science  and  an  art. 

13.  A  Problem  is  a  question  proposed  which  requires 
a  solution. 

13.  An  Operation  is  the  method  of  solving  a  problem. 

14.  An  Analysis  is  an  investigation  of  the  several 
parts  of  a  problem.  An  a^ialysis  leads  to  the  operation 
which  obtains  the  answer  to  a  problem. 

15.  A  Rule  is  a  direction  for  performing  an  opera- 
tion. A  rule  is  usually  derived  from  an  analysis  of  one 
or  more  problems. 


NOTATION.  11 


NOTATION. 


IG.  Notation  is  the  method  of  representing  num- 
bers by  figures  or  letters.  Arabic  notation  represents 
numbers  by  figures.  Roman  notation  represents  num- 
bers by  letters. 

ARABIC     NOTATION. 

PROBIiEM    I. 

17.  To  represent  an  integral  number  between  naught 
and  ten, 

Write  the  necessary  figure  as  in  Definition  6. 

EXAMPLE. 

1.  Write  in  figures,  three,  seven,  one,  five,  four,  eight, 
six,  nine,  two,  and  naught. 

2.  How  many  units  does  the  figure  5  represent  ? 

3.  How  many  marbles  does  the  figure  9  represent? 

4.  How  many  apples  have  you  when  they  can  be 
represented  by  0?     3?     4?     6?     7?     8?     2? 

PROBIiEM    II. 

18.  To  represent  any  integral  number  between  te7i 
and  nineteen,  inclusive. 

Place  the  several  figures  to  the  right  of  the  figure  1. 

EXAMPLE. 

1.  Write,  by  means  of  figures, 

ten    eleven   twelve   thirt'n    fonrt'n  fift'n  sixt'n  sevont'ii  eiglil'n   nitict'n 

Ans.    10    11      12      13      14    15    16      17       18      19 


12  NOTATION. 

The  figures  at  the  right  hand,  or  first  place,  in  these 
numbers  represent  units  of  the  first  order. 

The  figure  in  the  second  place  represents  a  unit  of 
THE  SECOND  ORDER. — (Vide  Def.  4.) 

PROBI.EM     III. 

lO.  To  represent  the  units  of  the  second  order, 
between  ten  and  ninety  inclusive, 

Place  the  cipher  at  the  right  of  the  several  digits. 

example. 
1.  Write,  by  the  aid  of  figures, 

ten,    twenty,    thirty,  forty,    fifty,     sixty,    seventy,    eighty,    ninety. 

Ans.  10     20       30    40    50     60      70       80       90 

PROBI.EM    IV. 

SO.  To  represent  any  integral  number  between 
twenty  and  one  hundred, 

Unite  the  required  units  of  the  first  order  to  those  of 
the  second. 

examples. 

1.  Write,  by  the  aid  of  figures,  the  number  twenty- 
five.  Here  five  units  of  the  first  order  are  required,  and 
two  of  the  second.  Ans.  25. 

2.  Write  on  your  slate  all  the  integral  numbers  be- 
tween naught  and  one  hundred,  by  the  aid  of  figures. 

PROBLEM    V. 

21.  To  represent  the  units  of  the  third  order, 
called  hundreds. 

Place  hvo  ciphers  to  the  right  of  the  several  digits, 
thus  : 


NOTATION.  13 

One  hundred... 100  Four  hundred, ..400  Seven  hundred. ..700 
Two  hundred. ..200  Five  hundred. ..500  Eight  hundred. ..800 
Three  hundred. 300         Six  hundred 600         Nine  hundred. ...900 

PROBLEM    TI. 

22.  To  represent  any  integral  number  between  one 
hundred  and  one  thousand, 

Write  down  the  figure  representing  the  required  units 
of  the  third  order,  and  annex  to  it  the  figures  repre- 
senting the  required  units  of  the  second  and  first  orders. 

EXAMPLES. 

1.  Write,  by  the  aid  of  figures,  one  hundred  and 
twenty-three.  Here  one  unit  of  the  third  order,  two  of 
the  second,  and  three  of  the  first,  are  required. 

Ans.  123, 

2.  Write,  by  the  aid  of  figures,  seven  hundred  and 
eight.  Here  seven  units  of  the  third  order,  none  of  the 
second,  and  eight  of  the  first,  are  required. 

Ans,  708. 

3.  Write  on  your  slate  five  hundred  and  forty-nine, 
three  hundred  and  forty,  six  hundred  and  one,  three 
hundred  and  seven,  nine  hundred  and  nine,  seven 
hundred  and  ten. 

4.  Write  all  the  integral  numbers  between  three 
hundred  sixty-five  and  four  hundred. 

5.  Write  all  the  integral  numbers  between  seven 
hundred  three  and  seven  hundred  fifty. 

PROBLEM     Til. 

23.  To   represent    units    of    the   fourth    order, 

called   THOUSANDS, 


J  4  NOTATION. 

Place  three  ciphers  to  the  right  of  the  several  digits^ 
thus: 

One  thousand 1000    Four  thousand. .4000    Seven  thousand. .7000 

Two  thousand '2000    Five  thousand..5000    Eight  thousand.. 8000 

Three  thousand. ..3000    Six  thousand. ..6000    Nine  thousand. ..9000 

Units  of  the  fifth  order  are  called  tens  of  thou- 
sands. 

Units  of  the  sixth  order  are  called  hundreds  of 
'Thousands. 

probxem  viii. 

24.  To  represent  any  number  of  thousands  by  the 
aid  of  figures, 

Write  the  figures  as  if  they  zvere  to  represent  a  number 
less  than  one  thousand,  and  then  annex  three  ciphers. 

EXAMPLES . 

1.  Write,  by  the  aid  of  figures,  one  hundred  and 
twenty-three  thousand.  Ans.  123000. 

2.  Write  seven  hundred  and  eight  thousand. 

A71S.  708000. 

3.  Write  five  hundred  and  forty-nine  thousand,  six 
hundred  and  one  thousand,  nine  hundred  and  ten  thou- 
sand, five  hundred  and  fifty-five  thousand,  twenty-one 
thousand. 

PROBIiEM    IX. 

25.  To  represent  any  integral  number  less  than  one 
million. 

Write  the  number  as  if  it  were  required  to  write  the 
thousands  only,  but  instead  of  the  three  ciphers  place  that 
part  of  the  number  less  than  a  thousand. 


NOTATION.  15 

EXAMPLES. 

1.  Write  one  hundred  and  twenty-three  thousand  one 
hundred  and  ninety-one.  Ans.  123191. 

2.  Write  seven  hundred  and  eight  thousand  two  hun- 
dred and  five.  Ans.  708205. 

3.  Write  nine  hundred  and  nine  thousand  nine  hun- 
dred and  nine.  Ans.  909909. 

4.  Write  four  hundred  and  forty-four  thousand  four 
hundred  and  forty-four.  Ans.  444444. 

5.  Write  twenty-seven  thousand  and  one. 

Ans.  27001. 

Remark. — Two  ciphers  are  retained  because  no  mention  is 
made  of   units  of  the  second  or  third  order. 

6.  Represent  in  figures  twenty-seven  thousand  three 
hundred  and  twenty-one,  twenty-seven,  three  hundred 
and  one,  twenty-seven  thousand  and  twenty-one,  tw^enty- 
seven  thousand  and  five,  sixty-seven. 

Units  of  the  seventh  order  are  called  millions. 

Units  of  the  eighth  order  are  called  tens  of  mill- 
ions. 

Units  op  the  ninth  order  are  called  hundreds  of 
millions. 

PROBI.£i!»I     X. 

26.  To  represent  in  figures  any  number  of  millions, 
Write  the  number  as  if  no  mention  were  made  of  mill- 
ions and  then  annex  six  ciphers. 

examples. 

1.  Write  one  hundred  and  twenty-three  millions. 

Ans.  123  000  000. 

2.  Write  seven  hundred  and  eight  millions. 

Ans.  708  000  000. 


10  NOTATION. 

27.  In  the  previous  section,  the  three  figures  on  the 
right  of  the  answer  to  each  example  form  the  period  op 

UNITS. 

The  three  next  figures  form  the  period  of  thousands. 
The  three  next  figures  form  the  period  of  millions. 
A  few  succeeding  periods  are  named  as  follows : 

4.  Billions.  7.  Quintillions.    10.  Octillions. 

5.  Trillions.  8.  Sextillions.      11.  Nonillions. 

6.  Quadrillions.     9.  Septillions.       12.  Decillions. 

Remark. — These  names  might  be  continued  to  any  extent. 
PROBIiEM     XI. 

28.  To  represent  in  figures  any  number  whatever, 
Write  each  period  in  the  order  named,  as  if  it  were  the 

period  of   units;    hut  if   intermediate  periods  are  not 
named,  fill  their  places  luifh  ciphers. 

exajiples. 

1.  Write  four  hundred  and  twenty-three  trillions  two 
hundred  and  seven  billions  five  hundred  and  six  thou- 
sand and  one.  Ans.  423,207,000,506,001. 

Remark. — For  convenience  the  periods  may  be  separated  by  a 
comma. 

2.  Write  in  figures  twenty-seven;  forty-three;  one 
hundred  and  fifty-four ;  five  thousand  and  ten ;  twenty- 
six  thousand  and  forty -five;  three  hundred  thousand 
and  seven ;  two  millions  and  five ;  forty-seven  millions 
and  thirty-three ;  three  decillions  one  nonillion  twenty- 
seven  octillions  three  quintillions  one  hundred  and 
twenty  billions  and  thirty-four. 

Last  Ans.  3,001,027,000,000,003,000,000,120,000,000,034. 


NUiMEllATION.  17 


NUMERATION 


29.  Numeration  exhibits  the  method  of  reading  num- 
bers represented  by  figures. 

EXA]\IPLES. 

1.  Read,  in  common  language,  the  number  631230- 
405078901. 

We  begin  by  separating  the  number  into  periods  of 
three  figures  each,  commencing  at  the  right.     Thus  : 
631,  230,  405,  078,  901. 

Next,  apply,  either  mentally  or  in  writing,  the  riames 
of  each  of  the  jjeriods  as  given  in  Notation,  beginning  at 
the  right.     Thus : 

Trill.      Bill.        Mill.      Tlions.    Units. 

631,  230,  405,  078,  901. 

Next,  hegin  at  the  left^  and  give  to  the  third  figure 
the  name  of  the  unit  of  the  third  order,  (vide  21,)  and 
to  the  second  figure  the  name  of  the  unit  of  the  second 
order,  (vide  19,)  and  to  the  first  figure  the  name  of  the 
digit  which  it  represents,  (vide  6.)  Finally,  apply  the 
name  of  the  period  itself. 

Do  the  same  thing  to  the  successive  periods  toward 
the  right,  omitting  the  names  of  places  filled  by  ciphers. 
Thus: 

Six  hundred  thirty-one  trillions  two  hundred  thirty 
billions  four  hundred  and  five  millions  seventy-eight 
thousand  nine  hundred  and  one. 


18  NUMERATIOX. 

Remark  1. — The  word  and  is  used  when  required  by  custom. 

Rkmark  2. — It  is  not  customary  to  apply  the  name  of  the  period 
on  the  right.  Nine  hundred  and  one  is  the  same  as  nine  hundred 
and  one  units. 

2.  Read  the  number  63021457823675301742601930. 


OPERATION. 
Scptil.  Sextil.     Quin.     Quad.     Trill.      Bill.       Mill.  Thoiis. 

63,  021,  457,  823,  675,  301,  742  601,  930. 
Sixty-three  septillions  twenty-one  sextillions,  etc. 

Remark. — It  is  observed  that  the  period  on  the  left  need  not 
be  full,  for  a  cipher  in  the  vacant  place  would  be  of  no  service. — 
(Vide  6.) 

3.  Read  the  number  319415012. 
Arts.  Three  hundred  nineteen  million  four  hundred 
fifteen  thousand  and  twelve. — (Vide  18.) 

30.  From  these  examples  we  have  the  following 

RULE  FOR  READING   NUMBERS. 

1.  Separate  the  number  into  periods  of  three  figures 
each,  beginning  at  the  right. 

2.  Apply  to  the  third,  second,  and  first  figures  of  the 
period  on  the  left,  if  it  is  fidl,  the  names  of  the  units  of 
the  third,  second,  and  first  orders,  and  afterivard  apply 
the  name  of  the  period,  omitting  names*  in  all  cases  where 
ciphers  occur. 

3.  Read  each  period  in  the  same  way,  passing  to  the 
right. 

4.  If  the  middle  figure  of  a  p)eriod  is  1,  apply  the  name 
which  it,  in  connection  ivith  the  figure  on  its  right,  rep- 
resents. 


NUMEllATION. 


19 


EXAMPLES. 


Read  the  following  numbers 


1. 

2. 
3* 
4. 
5. 
6. 


12. 

184. 

3261. 

81765. 

987123. 


7.  6345555.  14. 


9. 
10. 
11. 
12. 
13. 


3. 

10. 

530. 

4021. 

56304. 

740357. 


15. 
16. 
17. 

18. 
19. 
20. 


1312547. 

24340586. 

382003125. 

4960606544. 

51730004163. 

664800000532. 


6401836.  21.  7562345940051. 


22.  30014820016. 

23.  372536370001. 

24.  8134000500010. 

25.  370010050020. 

26.  1010101010101. 


27.  4040506070809. 

28.  3004005006001. 

29.  7000600050004. 

30.  9000040000700. 

31.  4000000130000. 


32.  Read  the  number  3014056000451. 

33.  Read  the  number  40700369997823. 

34.  Read  the  number  100370059431001. 

35.  Read  the  number  6001478900462357. 

36.  Read  the  number  81705430267891456. 

37.  Read  the  number  123456789098765432. 

38.  Read  the  number  9876543210123456789. 

39.  Read  the  number  10203040506070809011. 

40.  Read  the  number  908070605040302010999. 

41.  Read  the  number  1002003004005006007008. 

42.  Read  the  number  199:^991^91^Q1^g]^9^1. 


f 


20  DECIMALS. 


DECIMALS 


NOTATION   AND   NUMERATION. 
SI.  The  several  units  in  decimals  are  named  TENTH, 

HUNDREDTH,    THOUSANDTH,    TEN-THOUSANDTH,    HUNDRED- 
THOUSANDTH,    MILLIONTH,    etc. 

32.  The  relation  between  integral  and  decimal  units, 
and  the  manner  of  representing  them  by  figures,  are 
given  below. 

One  million 1000000. 

One  hundred  thousand 100000. 

One  ten  thousand 10000. 

One  thousand 1000. 

One  hundred 100. 

One  ten 10. 

One t One. 

.1 One  tenth. 

.01 One  hujidredth. 

.001 One  thousandth. 

.0001 One  ten-ihotisandth. 

.00001 One  hundred-thoiisandlh. 

.000001 One  millionth. 

(1.)  The  Decimal  Point  distinguishes  the  decimal  from 
the  in!9jfa0m^Sy^tsJpJa/e0>emg  at  the  left  of  all 
decimals.  The  period  (.)  is  commonly  employed  for  this 
purpose. 

(2.)  By  one  tenth  is  meant  one  of  the  ten  equal  parts 
into  which  the  nnit  one  is  divided. 


DECIMALS.  21 

(3.)  Bj  cue  hundredth  is  meant  one  of  the  hundred 
equal  parts  into  which  the  unit  one  is  divided. 
(4).  By  one  thousandth  is  meant,  etc. 

l>ROBL,£M    I. 

33.  To  represent  by  the  aid  of  figures  any  number 
of  tenths, 

Place  the  decimal  point  to  the  left  of  the  proper  digit. 

EXAMPLES. 

1.  Represent  five  tenths  with  a  figure.  Ans.  .5 

2.  Represent  two  tenths,  three  tenths,  four  tenths, 
six  tenths,  seven  tenths,  eight  tenths,  nine  tenths,  each 
by  the  proper  digit. 

Hemakk. — By  two  tenths  we  mean  two  of  the  ten  equal  parts  into 
which  the  unit  one  is  divided. 

PROBIi£M    II. 

34.  To  represent  by  the  aid  of  figures  any  number  of 
hundredths. 

(1.)  If  the  number  of  hundredths  is  less  than  ten, 

Place  a  cipher  between  the  decimal  point  and  the  proper 
digit. 

(2.)  If  the  number  of  hundredths  is  ten  or  more  than 
ten, 

Place  the  decimal  point  to  the  left  of  the  given  number. 

examples'. 

1.  Represent  two  hundredths  by  figures. 

Ans.  .02 

2.  Represent  three  hundredths,  four  hundredths,  etc., 
to  nine  hundredths  by  figures.  Ans.  .03   etc. 


22  DECIMALS. 

3.  Represent  ten  hundredths  by  figures. 

A71S,  .10 

4.  Represent  eleven  hundredths,  twelve  hundredths, 
etc.,  to  ninety-nine  hundredths.  Ans.  .11   etc. 

Remark. — By  tivo  hundredths  we  mean  two  of  the  hundred  equal 
parts  into  which  the  unit  one  is  divided. 

PROBIiEM    III. 

35.  To  represent  by  the  aid  of  figures  any  number  of 
thousandths. 

(1.)  If  the  number  of  thousandths  is  less  than  ten, 

Place  two  ciphers  between  the  decimal  point  and  the 
proper  digit. 

(2.)  If  the  number  of  thousandths  is  ten  or  more,  and 
less  than  a  hundred. 

Place  a  cipher  between  the  decimal  point  and  the  given 
number. 

(3.)  If  the  number  of  thousandths  is  one  hundred,  or 
more  than  one  hundred, 

Place  the  decimal  point  to  the  left  of  the  given  number. 

EXAMPLES. 

1.  Represent  three  thousandths  by  figures. 

Ans.  .003 

2.  Represent  one*  thousandth,  two  thousandths,  etc., 
to  nine  thousandths.  Ans.  to  last,  .009 

3.  Represent  ten  thousandths  by  figures. 

Ans.  .010 

4.  Represent  eleven  thousandths,  twelve  thousandths, 
etc.,  to  ninety-nine  thousandths.       Ans.  to  last,  .099 


DECIMALS.  23 

5.  Represent  one  hundred  thousandths. 

Ans,  .100 

6.  Represent  six  hundred  and  twenty-one  thou- 
sandths. Ans.  .621 

7.  Represent  three  hundred  and  six  thousandths. 

Ans.  .306 

8.  Represent  one  hundred  and  one  thousandths,  one 
hundred  and  two  thousandths,  etc. 

Ans.  .101   .102  etc. 

Remark. — By  /our  thousandlhs  we  mean /our  of  the  thousand  equal 
parts  into  which  the  icnit  one  is  divided. 

PROBLEM    IV. 

36.  To  represent  by  figures  any  decimal  whatever, 

(1.)    Write  the  given  number,  as  in  sections  17  to  28. 

(2.)  WJien  necessary,  prefix  ciphers  enough  to  make  the 
right-hand  figure  of  the  number  occupy  the  place  of  1 
when  representing  the  given  decimal  unit. — (Vide  32.) 

(3.)   To  the  left  place  the  decimal  point. 

Remark. — Observe  that  one  figure  only  on  the  right  of  the  point 
is  required  to  represent  tenths^  two  figures  to  represent  hundredths, 
three  figures  to  represent  thousandths,  etc.  (Vide  32.) 

EXAMPLES. 

1.  Represent  by  figures  one  ten-thousandth. 

Ans.  .0001 

2.  Represent  by  figures  two  ten-thousandtJis. 

Ans.  .0002 

3.  Represent  twenty-one  ten-thousandths. 

Ans.  .0021 

4.  Represent  three  hundred  and  six   ten-thousandths. 

Ans.  .0306 


24  DECIMALS. 

5.  Represent  three  thousand  and  five  ten-thousandths. 

Ans.  .3005 

6.  Represent  one  hundred-thousandth. 

Ans.  .00001 

7.  Represent  six  hundred  and  one  hundred-thou- 
sandths. Ans.  .00601 

8.  Represent  one  millionth.  Ans.  .000001 

9.  Represent  one  thousand  and  five  millionths. 

Ans.  .001005 

10.  Represent  bj  figures,  four  tenths^  twelve  hun- 
dredths, seven  hundredths,  five  thousandths,  thirty-seven 
thousandths,  one  hundred  and  eleven  thousandths,  forty- 
six  ten-thousandths,  nine  hundred  and  one  ten-thou- 
sandths, three  hundred  and  sixty-one  hundred-thou- 
sandths, ten  thousand  four  hundred  and  fifty-six 
millionths,  one  ten-millio7ith,  twenty-seven  ten-millionths, 
one  hundred-millionth,  sixty-five  hundred-millionths,  one 
billionth,  three  thousand  and  fifty-seven  hillionths. 

Ans.  to  last,  .000003057 

PR  OBI.  EM    V. 

•17.  To  read  any  decimal  represented  by  figures, 
(1.)    Read  the  figures  as  if  represe7iting  an  integral 

number. — (Vide  29.) 

(2.)   Apply  the  name  of  the  decimal  unit  indicated  by 

the  right-hand  figure. — (Vide  36,  Remark.) 

EXAMPLES. 

1.  Read  .5;  .05;  .005;  .0005;  .00005;  .000005  in 
words. 

2.  Read  .3;  .13;  .213;  .1111;  .22222;  .999999  in 
words. 


DECIMALS.  25 

3.  Read  .9;   .24;  .031;  .0461;   .00231;   .009999  in 

words. 

Remark. — An  integral  number  and  a  decimal  may  be  written 
together.     Thus :  3.7  are  three  and  seven  tenths. 

4.  Read  2.5;  4.05;  8.005;  21.0005  in  words. 

5.  Read  8.1;  2.12;  9.224;  27.1234  in  words. 

6.  Read  the  following  expressions  : 

3.004  23.005  67.431  5.6789 

17.115  48.673  12.6001  27.3004 

126.432  12.6432  1.26432  .126432 

.1345  1.345  13.45  134.5 

38.  Principles  of  Arabic  Notation. 
I.  All  numbers  are  derived  from  the  unit  one. 
II.  Removing  any  figure  one  place   toward  the  left 
increases  its  value  .ten  times.     Thus,  in  the  expressions 

.001  .01  .1  1.  10.  100. 

the  value  of  the  figure  1  is  increased  ten  times  in  each 
step  of  its  passage  from  right  to  left  past  the  decimal 
point. 

Remark  1. — Any  digit,  then,  may  have  a  simple  or  it  may  have 
a  local  value;  it  has  a  simple  value  when  written  alone,  and  a  local 
value  in  all  other  cases. 

m,       ^  r  The  simple  value  of  three  is  3. 

'  \  Some  local  values  of  three  are  30. ;  .3 ;  .03 

Remark  2. — The  principal  use  of  the  cipher  is  to  give  a  local 
value  to  the  digits. — (Vide  6.) 

Remark  3. — It  is  evident  that  one  tenth  of  any  quantity,  as  for 
instance  a  dollar,  is  the  same  as  ten  hundredths  of  the  same  quan- 
tity; that  is,  .1  is  the  same  as  .10;  hence. 

Placing  a  cipher  to  the  right  of  a  decimal,  does  not 
change  the  value  of  the  decimal. 
3 


26  DECIMALS. 

UNITED    STATES    MONEY. 

NOTATION  AND   NUMERATION. 

39.  The  several  units  of  the  currency  of  the  United 
States  are  named  the  Eagle,  Dollar,  Dime,  Cent,  and 
Mill.  Of  these  only  the  dollar,  cent,  and  mill  are  con- 
sidered in  arithmetical  Notation. 

40.  The  dollar  is  the  primary  unit,  and  figures  rep- 
resenting dollars  are  considered  as  integral  numbers. 

41.  The  cent  is  the  one  hundredth  part  of  one  dollar. 

42.  The  mill  is  the  one  thousandth  part  of  one  dollar. 

43.  The  sign  $,  when  placed  before  figures,  denotes 
that  United  States  money  is  meant. 

Remark. — The  gold  coins  of  the  United  States  are  the  double- 
eagle,  eagle,  half-eagle,  quarter-eagle,  and  dollar. 

The  silver  coins  are  the  dollar,  half-dollar,  quarter-dollar,  dime, 
and  half-dime.     The  nickel  coin  is  the  three-cent  piece. 

The  copper  coins  are  the  two-cent  and  the  one-cent  pieces.     ' 

The  value  of  the  eagle  is  ten  dollars,  and  of  the  dime  ten  cents. 

The  eagle  weighs  10  pennyweights  18  grains. 

PROBIiEM   I. 

44.  To  represent  by  figures  any  number  of  dollars, 
cents,  and  mills, 

(1.)  Place  the  figures  indicating  the  dollars  on  the  left 
of  the  decimal  point. 

(2.)  Consider  the  figures  indicating  the  cents  as  so 
many  hu7idredths,  arid  write  them  as  directed  by  34. 

(3.)  Write  the  figure  indicating  the  mills  in  the  third 
place  on  the  right  of  the  decimal  point. 

(4.)   To  the  whole  prefix  the  sign  |. 

Remark. — If  mills  only  are  to  be  represented,  the  places  of  the 
dollars  and  cents  must  be  filled  with  ciphers. — (Vide  35,  1.) 


DECIMALS.  27 

EXAMPLES. 

1.  Represent  one  mill  by  figures.        Arts,  f  0.001. 

2.  Represent  two  mills,  three  mills,  four  mills,  etc., 
to  nine  mills.  Ans.  to  last,  |0.009. 

3.  Represent  one  cent  by  figures.  Ans.  §0.01. 

4.  Represent  2  cents,  3  cents,  4  cents,  etc.,  to  9  cents. 

Ans.  to  last,  $0.09. 

5.  Represent  twenty-five  cents  by  figures. 

Ans.  10.25. 

6.  Represent  6  dollars  27  cents.  An^.  |6.27. 

7.  Represent  8  dollars  10  cents  4  mills.  * 

Ans.  $8,104. 

8.  Represent  24  dollars  25  cents  1  mill. 

Ans.  $24,251. 

9.  Represent  103  dollars  6  mills.    Ans.  $103,006. 

10.  Represent  904  dollars  37  cents. 

Ans.  $904.37. 

11.  Represent  1  dollar  20  cents  5  mills. 

Ans.  $1,205. 

12.  Represent  4  cents;  5  mills;  13  cents;  65  cents; 
5  dollars  14  cents  2  mills;  167  dollars  55  cents  7  mills; 
1  dollar  1  mill;  65  dollars  6  mills;  125  dollars. 

Ans.  to  last,  $125.00. 

13.  Represent  4  dollars  3  cents  1  mill;  6000  dollars 
1  cent;  1245  dollars  3  mills;  45  dollars;  7  mills;  5  cents; 
222  dollars  22  cents  2  mills;  9167  dollars  54  cents  9 
mills. 

14.  What  is  meant  by  the  expression  $1,251  ? 

Alls.  1  dollar  25  cents  1  mill. 

15.  Read  in  words  the  following  sums  of  money : 


28 


DECI3: 

[ALS. 

13.265 

§3.043 

$20,072 

$0,714 

19.118 

§6.259 

$18,013 

$7.14 

$0,001 

$0,141 

$00,162 

$71.40 

$1,111 

$5,001 

$11,001 

$714. 

45.   Exercises  ix  Review. 

1.  Express  by  figures  twenty-six;  one  hundred  and 
one ;  six  hundred  and  forty ;  seven  hundred  and  fifty- 
three  ;  five  hundred  and  sixty-seven ;  three  hundred  and 
eleven;  six  thousand  and  four;  eight  thousand  and 
ninety. 

2.  Expre'ss  by  figures  two  thousand  and  twenty-two ; 
three  million  and  twenty-two;  forty -five  million  and 
twenty-two ;  three  hundred  and  eight  thousand. 

3.  Express  by  figures  six  billion  and  five  million; 
eight  trillion  and  seven  thousand;  six  quadrillion  and 
one ;  three  quintillion  and  sixty-seven. 

4.  Express  by  figures  four  hundred  and  forty-three 
thousand  five  hundred  and  twenty-five. 

5.  Express  by  figures  sixty-eight  billion  two  hundred 
and  three  million  five  hundred  and  five  thousand  six 
hundred  and  forty-five. 

6.  Express  by  figures  twenty-six  Jiundredths;  one 
hundred  and  one  thousandths;  six  hundred  and  forty 
millionths;  sixty-four  hundred-thousandths. 

7.  Express  by  figures  seven  tenths ;  seven  hundredths ; 
seven  thousandths ;  sixty-five  thousandths;  one  hillmith; 
one  trillionth;  one  quadrillionth;  one  quintilliontli ;  one 
sextillionth;  one  septilUonth;  three  octilllonths;  seven 
nonillionths;  one  hundred  and  eleven  deeilliontlis. 

8.  The  number  of  inches  from  the  Equator  to  the 


DECIMALS.  29 

North  Pole  is  three  hundred  ninety-three  million  seven 
hundred  seven  thousand  nine  hundred.  What  figures 
express  them? 

9.  The  numher  of  seconds  in  a  year  is  thirty-one 
million  five  hundred  fifty-six  thousand  nine  hundred 
twenty-seven  and  fifty-seven  hundredths.  What  figures 
express  them  ? 

10.  The  distance  from  the  earth  to  the  moon  is  two 
hundred  thirty-eight  thousand  six  hundred  and  fifty 
miles.     What  figures  express  this  distance  ? 

11.  The  distance  from  the  earth  to  the  sun  is  about 
ninety-five  million  miles.  What  figures  express  this 
distance  ? 

12.  It  is  about  twenty  trillion  miles  to  the  nearest 
star.     Express  the  distance  in  figures. 

13.  Express  by  the  aid  of  figures  two  dollars  sixteen 
cents  four  mills ;  six  dollars  seven  cents ;  eight  dollars 
and  one  mill;  one  hundred  twenty-five  dollars  sixty 
cents.  Ans.  to  last,  §125.60 

14.  Express  by  the  aid  of  figures  one  hundred 
twenty-five ;  one  hundred  twenty -five  thousandths ;  one 
dollar  twenty -five  cents ;  twelve  dollars  fifty  cents ; 
twelve  cents  five  mills ;  one  and  twenty-five  hundredths ; 
twelve  and  five  tenths;  twelve  and  fifty  hundredths. 

15.  Read  the  following  expressions:  3071;  $3071; 
3.071;  §3.071;  30.71;  §30.71;  307.1;  §307.10; 
307.10;  307.100. 


30  ROMAN   NOTATION   AND   NUMERATION. 


ROMAN  NOTATION  AND  NUMERATION 


46.  The  Roman  Notation  makes  use  of  seven  Capi- 
tal Letters  to  represent  numbers.     They  are 

I,        V,        X,         L,         C,         D,         M; 
and  their  values  are,  respectively, 

1,         5,        10,        50,       100,      500,      1000. 

47.  Principles  of  Roman  Notation. 

I.  The  repetition  of  a  letter  repeats  the  value  of  the 
letter.  Thus :  II  are  2,  III  are  3,  XX  are  20,  XXX  are 
30,  CCC  are  300. 

II.  If  a  letter  is  placed  hefore  another  of  greater  value 
than  itself,  the  value  of  the  less  is  taken  from  that  of  the 
greater.  Thus :  IV  represent  4,  XL  represent  40,  XC 
represent  90. 

III.  If  a  letter  is  placed  after  another  of  greater  value 
than  itself,  and  a  letter  of  greater  value  does  not  follow 
both  of  them,  the  value  of  the  less  is  added  to  that  of 
the  greater.  Thus :  XI  represent  11,  XIV  represent  14, 
OX  represent  110,  CXL  represent  140. 

48.   Examples. 

1.  Represent  by  the  aid  of  letters  the  numbers  1,  2, 
3,  4,  5.  Ans.  I,  II,  III,  IV,  V. 

2.  Represent  by  the  aid  of  letters  the  numbers,  6,  7, 
8,  9, 10.  A71S.  VI,  VII,  VIII,  IX,  X. 


ADDITIOX.  31 

3.  Represent  by  the  aid  of  letters  the  numbers  11, 
12,  13,  etc.,  to  50.  Ans.  XI,  XII,  XIII,  etc.,  L. 

4.  Represent  54,  80,  90,  100,  150,  199,  500,  1099, 
1865.  Ans.  to  List,  MDCCCLXV. 

5.  Represent  60,  63,  71,  94,  83,  101,  565,  1741. 

Ans.  to  last,  MDCCXLI. 

6.  Represent  1001,  1005,  1008,  1010,  1499. 

Ans.  to  last,  MCDXCIX. 

7.  Represent  1410,  1951,  1673,  1467,  1866. 

Ans.  to  last,  MDCCCLXVI. 


ADDITION 


49.  Addition  is  the  operation  of  finding  the  sum  of 
two  or  more  numbers. 

50.  The  sum  is  a  number  which  contains  as  many 
units  as  all  the  numbers  taken  together.  Thus  :  the  sum 
of  5  and  3  is  8. 

•    SIGNS. 

51.  The  sign  +  is  called  plus,  and  signifies  that  the 
numbers  between  which  it  is  placed  are  to  be  added  to- 
gether.    Plus  is  a  Latin  word,  signifying  more. 

52.  The  sign  =  is  called  the  sign  of  equality,  and 
signifies  that  the  lohole  expression  placed  before  it  is 
equal  to  that  placed  after  it.  Thus:  5-f  3=8,  is  read 
five  plus  three  equals  eight,  and  the  meaning  is  that 
there  are  the  same  number  of  units  in  8  as  in  5  and 
3  taken  together. 


32 


ADDITION. 


53.  In  the  following  table  the  sign  +  may  be  read 
by  the  word  and,  the  sign  =  by  the  word  are.  Thus : 
5  and  3  are  8. 


ADDITIO 

N     TABLE. 

2  +  0=2 

3  +  0=3 

4  +  0=4 

5  +  0=5 

2  +  1=3 

3  +  1=4 

4  +  1=5 

5  +  1  =    6 

2  +  2=4 

3  +  2=5 

4  +  2  =    6 

5  +  2=7 

2  +  3=5 

3  +  3=6 

4  +  3=7 

5  +  3=8 

2  +  4=6 

3  +  4=7 

4+4=8 

5  +  4=9 

2  +  5=7 

3  +  5=8 

4  +  5=9 

5  +  5  =  10 

2  +  6=8 

3  +  6=9 

4  +  6  =  10 

5  +  6  =  11 

2  +  7=9 

3  +  7  =  10 

4  +  7  =  11 

5  +  7  =  12 

2  +  8  =  10 

3  +  8  =  11 

4  +  8  =  12 

5  +  8  =  13 

2  +  9  =  11 

3  +  9  =  12 

4  +  9  =  13 

5  +  9  =  14 

6  +  0  =    6 

7  +  0=7 

8  +  0=8 

9  +  0=9 

6  +  1=7 

7  +  1=8 

8  +  1  =     9 

9  +  1  =  10 

6  +  2=8 

7  +  2=9 

8  +  2  =  10 

9  +  2  =  11 

6  +  3  =    9 

7  +  3  =  10 

8  +  3  =  11 

9  +  3  =  12 

6  +  4  =  10 

7  +  4.=  11 

8  +  4  =  12 

9  +  4  =  13 

6  +  5  =  11 

7  +  5  =  12 

8  +  5  =  13 

9  +  5  =  14 

6  +  6  =  12 

7  +  6  =  13 

8  +  6  =  14 

9  +  6  =  15 

6  +  7  =  13 

7  +  7  =  14 

8  +  7  =  15 

9  +  7  =  16 

6  +  8  =  14 

7  +  8  =  15 

8  +  8  =  16 

9  +  8  =  17 

6  +  9  =  15 

7  +  9  =  16 

8  +  9  =  17 

9  +  9  =  18 

PROBLEM    I. 

54.  To  add  any  number  of  figures  representing  units 
of  the^rs^  order,  (vide  18,) 

(1.)  Set  the  figures  under  each  other,  and  add  from  the 
bottom  upward  or  from  the  top  downward. 

(2.)  Place  the  sum  under  the  column,  so  that  the  figure 
representing  units  of  the  first  order  shall  fall  exactly/ 
underneath  the  figures  above. 


AJJi 

UllUiN. 

i5 

EXAMPLES. 

(1.)   (2.) 

(3.) 

(4.)   (5.) 

(6.)   (7.) 

(8.)   (9.) 

(10.) 

4       5 

1 

6       8 

7      5 

2       4 

5 

6       8 

0 

7      2 

2       9 

0       3 

9 

3       2 

7 

2       9 

4       8 

0       2 

6 

7       9 

3 

5       4 

3      1 

9       5 

7 

20     24 

11 

20     23 

16     23 

11     14 

27 

33 


11.  Add  6,  3,  5,  9,  4,  0,  2.  15.  Add  9,  8,  7,  6,  5,  4,  3,  2,  1. 

12.  Add  8,  3,  9,  0,  9,  9,  9.  16.  Add  1,  4,  7,  2,  5,  8,  3,  6,  9. 

13.  Add  4,  1,  3,  2,  5,  7,  9.  17.  Add  3,  2,  5,  7,  9, 1,  4,  6,  8. 

14.  Add  8,  2,  6,  4,  1,  5,  0.  18.  Add  1,  3,  5,  7,  9,  2,  4,  6,  8. 

PR  OB  I.  EM    II. 

55.  To  add  any  number  of  figures  representing  units 
of  the  second  order,  (vide  19,) 

(1.)  aS'^^  the  figures  under  each  other,  and  add  as  in 
54,  (1.) 

(2.)  Place  the  sum  under  columns,  so  that  the  figure 
represeyiting  units  of  the  second  order  shall  fall  exactly 
underneath  the  digits  above. 


EXAMPLES 

. 

(1.) 

(2.) 

(3.) 

(4.)     (5.) 

(6.) 

(7.) 

(8.) 

40 

50 

10 

90       50 

50 

80 

70 

60 

80 

00 

80       40 

90 

50 

30 

30 

20 

70 

70       70 

80 

30 

20 

70 

90 

30 

60       30 

70 

40 

40 

200   240  110   300  190  290   200  160 
9.  Add  60,  30,  50,  90,  40.   12.  Add  10,  20,  90,  80. 

10.  Add  10,  20,  40,  70,  80.   13.  Add  30,  50,  70,  90. 

11.  Add  60,  50,  40,  30,  20.   14.  Add  20,  40,  60,  80. 


34 


ADDITION. 


Remark. — It  is  evident  that  figures,  all  of  which  represent  any 
one  given  order  of  units,  may  be  added  in  the  same  way. 


(15.) 
400 

600 
300 
700 


(16.) 
5000 
8000 
2000 
9000 


(17.) 
10000 
00000 
70000 
30000 


(18.)  (19.) 

900000  6000000 

800000  7000000 

700000  1000000 

600000  9000000 


2000  24000         110000       3000000  23000000 

In  the  same  manner  add — 

(20.)  (21.)  (22.)  (23.) 

8  70  500  4000 

6  20  400  3000 

5  70  800  9000 


19  160  1700 

24.  Add  4,  9,  3,  7,  6. 

25.  Add  5,  8,  4,  3,  0,  7, 1. 

26.  Add  30,  40,  50,  60. 

27.  Add  400,  300,  500,  100. 

28.  Add  9000,  4000,  3000. 

29.  Add  40000,  30000,  10000. 


16000 

Ans.  29. 

Ans.  28. 

Ans.  180. 

Ans.  1300. 

Ans.  16000. 

Ans.  80000. 


l»ROBI.EM    III. 

56.  To  add  any  numbers  together  where  the  sum  of 
the  corresponding  orders  of  units  in  all  the  numbers  is 
9  or  less  than  9, 

(1.)  Write  the  numbers  so  that  the  corresponding  orders 
of  units  may  stand  mider  each  other. 

(2.)  Begin  at  the  right,  and  add  each  column  separately, 
placing  the  smn  exactly  under  the  column  added. 


ADDITION. 


35 


EXAMPLES. 

1.  Add  together  19,  160,  1700,  and  16000. 

OPERATION. 

19 
160 

1700 
16000 


Ans.  9999. 
Ans.  8778 


17879  Ans. 

2.  Add  1025,  6712,  1111,  1151. 

3.  Add  1234,  4321,  2222,  1001. 

4.  Add  31004, 13121,  22102,  21101,  11210. 

Ans.  98538 

5.  Add  9, 10,  300, 130,  4110,  71100.     Ans.  75659 

PROBIiEMIV. 

57.  To  add  any  numbers  whatever  together. 

EXAMPLES. 

1.  Add  together  4578,  3426,  and  9875. 


OPERATION. 

4578 

•  3426 

9875 

Vide  55,  Ex. 

20  . 

.  .   19 

Vide  55,  Ex. 

21  . 

.  .  160 

Vide  55,  Ex. 

22  . 

.  .  1700 

Vide  55,  Ex. 

23  . 
1  . 

.  16000 

Vide  56,  Ex. 

.  nS79  Ans 

36  ADDITION. 

A  moment's  attention  shows  how  the  above  operation 
may  be  contracted. 

The  sum  of  the  first  column  is  19,  which  is  composed 
of  9  units  of  the  first  order  and  1  of  the  second.  Set 
down  the  9  units  under  the  units  of  the  first  order,  j^r  ^o 
and  add  the  1  unit  of  the  second  order  to  the  342(3 
column  of  units  of  the  same  order,  making  17,  9875 
which  is  composed  of  7  units  of  the  second  order,  TI^Z^ 
and  1  of  the  third.  Set  down  the  7  units  of  the 
second  order  under  that  column,  and  add  the  1  unit  of 
the  third  order  to  the  column  of  units  of  that  order, 
making  18,  which  is  composed  of  8  units  of  the  third 
order  and  1  of  the  fourth.  Set  down  the  8  units  of  the 
third  order  under  that  column,  and  add  the  1  unit  of  the 
fourth  order  to  the  column  of  units  of  that  order,  making 
17,  which  is  written  down  as  in  55.     Hence, 

RULE. 

(1.)  W^Hte  the  numbers  so  as  to  j^l^toe  the  figures  in  the 
corresponding  orders  of  units  directly  under  each  other, 
and  draw  a  line  underneath. 

(2.)  Begin  at  the  right  hand,  and  add  each  column 
separately,  setting  doivn  the  right-hand  figure  of  the  result 
under  the  column  added,  and  add-  the  left-hand  figure  or 
figures  to  the  next  column  on  the  left. 

(3.)  Set  dozvn  the  tuhole  amount  of  the  last  column. 

2.  Add  234,  589,  613.  Ans.  1436. 

3.  Add  7123,  6054,  and  9123.      Ans.  22300. 

4.  Add  70561,  23564,  and  34625.  Ans.  128750. 

5.  Add  123456,  654321,  456123.  Ans.  1233900. 

6.  Add  123,  240,  85,  36,  and  7.       Ans.  491. 


ADDITION. 


7.  Add  1,  7,  43,  76,  65,  15,  and  100.     Ans.  307. 

8.  Add  13,  165,  48,  6251,  and  19.        A71S.  6496. 

9.  Add  108,  5012,  4103,  60450,  and  6. 

Ans.  69679. 

10.  Add  3456,  6543,  4563,  3645,  and  5634. 

A71S.  23841. 

11.  Add  31236,  415,  621437,  90053,  and  34. 

Ans.  743175. 

12.  Add  31,  280,  4560,  78930,  and  672140. 

Ans.  755941. 


(13.) 

(14.) 

(15.) 

12343247 

213673 

13021654 

6015400 

13021654 

12343247 

13021654 

6015400 

6015400 

213673 

12343247 

213673 

31593974 

(16.) 

(17.) 

12346721305 

3126754 

8917259679 

25678960 

763421893 

763421893 

25678960 

8917259679 

3126754 

12346721305 

22056208591 

58.  To  add  several  decimals  together,  proceed  exactly 
as  in  57,  and  then  place  the  decimal  p>oint  in  the  sum 
directly  under  the  decimal  points  above. — (Yide  38,  Re- 
mark 3.) 


8 

ADDITION. 

EXAMPLES. 

(1.) 

(2.) 

(3.) 

(4.) 

^34 

71.23 

705.61 

314.5 

.589 

60.54 

235.64 

21.346 

.613 

91.23 

*      346.25 

5.17 

1.436 

223.00 

1287.50 

341.016 

(5.) 

(6.) 

(7.) 

345.012 

785.432    • 

987.65 

45.78 

1234.6 

12.1453 

121.3 

257.87 

1.67 

87.125 

2 

12.431 

1436.123 

599.217 

1290.333 

2437.5883 

8.  Add  12.4,  3.47,  27.67,  and  86.       Ans.  129.54. 

9.  Add  1.24,  34.7,  2.767,  and  .86.     Ans.  39.567. 

10.  Add  .124,  .347,  .2767,  and  8.6.      Ans.  9.3477. 

11.  Add  57.76,  98.54,  38.72,  and  43.65. 

Ans.  238.67. 

12.  Add  5.776,  985.4,  38.72,  and  4365. 

Ans.  5394.896. 

13.  Add  577.6,  9.854,  3.872,  and  .4365. 

Ans.  591.7625. 

14.  Add  4.8,  43.31,  74.019,  and  11.204. 

Ans.  133.333. 

15.  Add  29.0029,  3.4476,  and  58.123. 

Ans.  90.5735. 

16.  Add  twelve  and  four  tenths,  three  and  forty-seven 
hundredths,  twenty-seven  and  sixty-seven  hundredths, 
and  eighty-six.  Ans.  129.54. 

17.  Add    twenty-nine   and   twenty-nine      ten-thou- 


» 


ADDITIOX.  39 

sandths,  three  and  four  thousand  four  hundred  and 
seventy-six  ten-thousandths,  fifty-eight  and  one  hun- 
dred and  twenty-three  thousandths.        Ans.  90.5735. 

18.  Add  three  and  seven  tenths,  four  and  five  hun- 
dredths, one  hundred,  five  thousandths,  sixty-seven 
millionths,  five  hundred  and  three,  eight  and  six  ten- 
thousandths.  Ans.  618.755667. 

59.  To  add  United  States  Money,  consider  the  several 
items  as  decimals,  adding  as  in  58 ;  then  prefix  the  sign 
%  to  the  sum.— (Vide  44.) 

EXAMPLES. 

1.  Add  two  dolhirs  sixteen  cents  four  mills,  six  dol- 
lars seven  cents,  eight  dollars  one  mill,  one  hundred 
twenty-five  dollars  and  sixty  cents. 

OPERATION. 

_  $2,164 
6.07 
8.001  . 
125.60 

§141.835  Ans. 

2.  Add  §241.075,  $45.06,  |37.05,  §1216.131. 

Ans.  §1539.316. 

3.  Add  §3124.162,  §812.95,  §67.12,  §2145.75. 

A71S.  §6149.982. 

4.  Add  §1.132,  §56.075,  §931.87,  §4621.953. 

Alls.  §5611.03. 

5.  Add  §27.413,  §45.084,  §607.219,  §205.03,  §25.25, 
and  §405.006.  Ans.  §1315.002. 


40  ADDITION. 

6.  Add  $136,255,  §10.30,  $248.50,  $100,125,  and 
$65.38.  Ans.  $560.56. 

7.  Add    $2600,   $1927.404,   $1603.40,   $3304.17, 
$165.47,  and  $2600.08.  Ans,  $12200.524. 

8.  Add  $170,  $400.02,  $130,  $250.10,  and  $845.22. 

Ans.  $1795.34. 
9    Add  $17.15,  $23.43,  $7.19,  $8.37,  and  $12,315. 

Ans.  $68,455. 

10.  Add  $6.75,  $2.30,  $0.92,  $0,125,  and  $0.06. 

Ans.  $10,155. 

11.  Add  $56.18,  $7,375,  $280.00,   $0,287,   $17.00, 
and  $90,413.  A71S.  $451,255. 

12.  Add  241  dollars  7  cents  5  mills,  45  dollars  6  cents, 
37  dollars  5  cents,  and  1216  dollars  13  cents  1  mill. 

Ans.  $1539.316. 

Remark. — In  adding  a  long  column  of  figures,  it  is  of  much 
assistance  to  divide  it  into  several  parts  at  pleasure,  add  each  of 
the  parts  separately,  and  finally  the  several  partial  sums  for  the 
sum  total. 


(13.) 

(14.) 

(15.) 

45678 

76.345 

$27,251 

12345 

18.237 

43.026 

37425 

5.404 

126.007 

3128-  98576 

12.36  -112.346 

185.214 

8462 

1.1 

243.671 

71351 

33.33 

453.172-1078.341 

81250 

45.54 

999.999 

11111-172174 

8.8  -  88.77 

471.862 

3333 

75.464 

125.281 

7812 

21.853 

931.452 

4512 

27.306 

813.161 

76251-  91908 

31.452-156.075 

13.20  -3354.955 

362658 

357.191 

$4433.296 

ADDITION. 

41 

(16.) 

(17.) 

(18.) 

43267 

143.01 

$25.04 

14567 

26.435 

87.05 

76543 

506.146 

125.113 

81234 

81.237 

37.40 

30506 

67.21  - 

-  824.038 

103.046 

4736- 

-250853 

1.004 

95.062 

154 

65.042 

127.111 

58463 

121.251 

1237.086 

81460 

67.132 

906.07  - 

-2742.978 

70120 

9.25  - 

-  263.679 

81.023 

93126 

14.062 

3410.192 

47615- 

-350938 

87.643 

1.20 

82361 

100.916 

19.02 

' 

95864 

2147.05  - 

-2349.671 

127.45 

3729 

432.876 

87.40  - 

-3726.285 

26- 

-181980 

91.91 

487.103 

9428 

125.125 

45.073 

32193 

37.126 

110.029 

86159- 

-127780 

85.437- 

-  772.474 

$ 

3145.671- 

-3787.876 

911551 

4209.862 

10257.139 

PRACTICAL     EXAMPLES. 

19.  A  gentleman  purchased  234  bushels  of  corn  at 
one  time,  589  at  another,  and  613  at  another.  How 
many  bushels  did  he  buy  in  all? — (Vide  57,  Ex.  2.) 

20.  During  one  year  my  crop  of  cotton  was  sold  for 
$7123.00 ;  the  next  year  it  brought  $6054,  and  the  year 
after  I  received  |9123.00.  How  much  did  I  receive  for 
cotton  during  the  three  years? — (Vide  57,  Ex.  3.) 

21.  January  has  31  days,  February  28,  March  31, 

4 


42  ADDITION. 

April  30,  May  31,  June  30,  July  31,  August  31,  Sep- 
tember 30,  October  31,  Noveraber  30,  December  31. 
How  many  days  in  the  year?  Ans.  365. 

22.  Washington  was  born  in  1732  and  lived  67  years. 
In  what  year  did  he  die?  Ans.  1799. 

23.  From  the  creation  of  the  world  to  the  flood,  there 
were  1656  years ;  from  the  flood  to  the  siege  of  Troy, 
1164  years;  from  the  siege  of  Troy  to  the  building  of 
Solomon's  Temple,  180  years;  from  the  building  of  the 
Temple  to  the  birth  of  Christ,  1004  years.  In  what 
year  of  the  world  did  the  Christian  Era  commence  ? 

Ans.  4004. 

24.  How  many  years  have  intervened  from  the 
creation  of  the  world  to  the  year  1865?      Ans.  5869. 

25.  Homer  was  born  733  years  before  the  Christian 
Era.  How  many  years  from  the  birth  of  Homer  to  the 
year  1865  ?  Ans.  2598. 

26.  I  bought  a  barrel  of  flour  for  |6.78;  ten  pounds 
of  raisins  for  |2.30;  seven  pounds  of  sugar  for  |0.92; 
one  pound  of  coWeQ  for  $0,125,  and  two  oranges  for 
10.10.     What  was  the  whole  amount?    Ans.  |10.225. 

27.  A  collector  has  bills  in  his  possession  of  the  fol- 
lowing amounts:  one  of  §43.75;  another  of  $29.18; 
another  of  $17.63;  another  of  $268.95,  and  anotlicr  of 
$718.07.     What  amount  has  he  to  collect? 

Ans.  $1077.58 

28.  A  man  has  the  following  sums  of  money  due  him, 
viz:  $420,197,  $105.50,  $304,005,  $888,455.  What  is 
the  amount  due  him?  Ans.  $1718.157.  . 

29.  What  is  the.  sum  of  429,  21.37,  355.003,  1.07, 
and  1.7?  Ai^s.  808.143. 


ADDITIOX.  43 

30.  What  is  the  sum  of  .2,  .80,  .089,  .006,  .9000,  and 
.005?  Ans.2. 

31.  A  gentleman  bought  at  one  thne  13.25  bushels  of 
corn;  at  another,  8.4  bushels;  at  another,  23.051 
bushels ;  at  another,  6.75  bushels.  How  many  bushels 
did  he  buy  in  all?  Ans.  51.451  bushels. 

32.  A  gentleman  owns  five  farms ;  the  first  is  worth 
§11500;  the  second,  §3057;  the  third,  §2468;  the 
fourth,  §9462;  and  the  fifth  is  worth  as  much  as  the 
four  together.     What  is  the  value  of  the  five  farms  ? 

Ans.  §52974. 

33.  By  the  census  of  1850,  the  population  of  the  ten 
largest  cities  of  the  United  States  was  as  follows :  New 
York,  515547 ;  Philadelphia,  340045 ;  Baltimore,  169054 ; 
Boston,  136881;  New  Orleans,  116375;  Cincinnati, 
115436;  Brooklyn,  96838;  St.  Louis,  77860;  Albany, 
50763;  Pittsburg,  46601.  What  was  the  population  of 
all  combined?  Ans.  1665400. 

34.  By  the  census  of  1860,  the  population  of  the 
following  cities  was  ascertained  to  be — of  New  York, 
805651;  Philadelphia,  562529;  Brooklyn,  266661; 
Baltimore,  212418;  Boston,  •  177812;  New  Orleans, 
168675;  St.  Louis,  160773;  Cincinnati,  161044;  Chi- 
cago, 109260;  Bufi-alo,  81129;  Louisville,  68033 ;  New- 
ark, 71914;  San  Francisco,  56802;  Washington,  61122; 
Providence,  50666;  Rochester,  48204;  Detroit,  45619; 
Milwaukee,  45246;  Cleveland,  43417;  Charleston, 
40578;  Troy,  39232 ;  New  Haven,  39267;  Richmond, 
37910;  Lowell,  36827;  Mobile,  29258;  Jersey  City, 
29226;  Portland,  26341 ;  Cambridge,  26060;  Roxbury, 
25137;  Charlestown,  25063;  Worcester,  24960;  Utica, 


44 


ADDITION. 


22529;  Reading,  23161;  Salem,  22252;  New  Bedford, 
22309;  Dayton,  20081;  Nashville,  16988.  How  many 
inhabitants  in  all  these  cities  combined?  Ans. 

35.  By  the   census  of  1860,  the  population  of  the 
several  States  and  Territories  was  as  follows : 


Alabama 964201 

Arkansas 435450 

California 379994 

Connecticut 460147 

Delaware 112216 

Florida 140425 

Georgia 1057286 

Illinois 1711951 

Indiana 1350428 

Iowa 674948 

Kentucky 1155684 

Louisiana 708002 

Maine 628279 

Maryland 687049 

Massachusetts 1231066 

Michigan 749113 

Minnesota 173855 

Mississippi 791305 

Missouri 1182012 

New  Hampshire 326073 

New  Jersey 672035 


New  York 3880785 

North  Carolina 992G22 

Ohio 2339502 

Oregon 52465 

Pennsylvania 2906115 

Rhode  Island 174620 

South  Carolina 703708 

Tennessee 1109801 

Texas 604215 

Vermont 315098 

Virginia 1596318 

Wisconsin 775881 

Colorado 34277 

Dakotah ,.       4837 

District  of  Columbia...     75080 

Kansas 107206 

Nebraska 28841 

New  Mexico 93516 

Utah.." 40273 

Washington 11594 

Nevada 6857 


How  many  inhabitants  in  the  United  States  in  1860? 

Ans.  31445080. 

36.  How  many  inhabitants  in  the  six  New  England 
States  taken  together?  Ans.  3135283. 

37.  How  many  inhabitants  in  the  States  bordering  on 
the  Gulf  of  Mexico?  Ans.  3208148. 

38.  How  many  inhabitants  in  the  States  watered  bj 
the  Tennessee  River?  Ans.  4020991. 


SUBTRACTIOX.'  45 

39.  How  many  inhabitants  in  the  States  bounded  in 
part  by  the  Ohio  River?  Ans.  8153883. 

40.  How  many  inhabitants  in  the  States  watered  by 
the  Mississippi  River?  Ans.  8718889. 

41.  How  many  inhabitants  in  the  States  and  Terri- 
tories lying  wholly  west  of  the  Mississippi  River? 

Ans.  3830340. 


SUBTRACTION 


60.  Subtraction  is  the  operation  of  finding  the  dif- 
ference between  two  numbers. 

61.  The  difference  is  such  a  number  as  added  to  the 

less  will  give  the  greater. 

SIG-NS. 

63.  The  sign  —  is  called  minus,  and  when  placed 
between  two  numbers  it  signifies  that  the  one  on  the 
right  is  to  be  subtracted  from  that  on  the  left.  3Iinus 
is  a  Latin  word,  signifying  less. 

63.  The  expression  8 — 5=3,  is  read  eight  minus  five 
EQUALS  three,  and  the  meaning  is,  that  three  is  the  dif- 
ference between  eight  and  five.  The  expression  may 
also  be  read,  five  from  eight  are  three. 

64.  The  greater  of  the  two  numbers  is  called  the 
minuend,  and  the  smaller  is  called  the  subtrahend.  The 
result  of  the  subtraction  is  called  the  difference,  and 
oftentimes  the  reniainde''\ 


46 


SUBTIl  ACTION. 
SUBTRACTION     TABLE. 


2  —  2  =  0 

3-3  =  0 

4  —  4  =  0 

5-5  =  0 

3-2  =  1 

4-3  =  1 

5-4=1 

6-5  =  1 

4  —  2  =  2 

5-3  =  2 

6—4  =  2 

7  —  5  =  2 

5-2  =  3 

6-3  =  3 

7-4  =  3 

8-5  =  3 

6-2  =  4 

7-3  =  4 

8  —  4  =  4 

9  —  5  =  4 

7-2  =  5 

8-3  =  5 

9-4  =  5 

10  -  5  =  5 

8  —  2  =  6 

9-3  =  6 

10  -  4  =  6 

11  —  5  =  6 

9-2  =  7 

10  —  3  =  7 

11  _  4  =  7 

12  —  5  =  7 

10  —  2  =  8 

11  -  3  =  8 

12  -  4  =  8 

13  -  5  =  8 

11  -  2  =  9 

12  —  3  =  9 

13  —  4  =  9 

14  —  5  =  9 

6-6  =  0 

7-7  =  0 

8  —  8  =  0 

9-9  =  0 

7-6  =  1 

8-7  =  1 

9-8=1 

10  —  9  =  1 

8-6  =  2 

9  —  7  =  2 

10  —  8  =  2 

11  -  9  =  2 

9-6  =  3 

10  -  7  =  3 

11  -^  8  =  3 

12  -  9  =  3 

10  —  6  =  4 

11  -  7  =  4 

12  -  8  =  4 

13  —  9  =  4 

11  -  6  =  5 

12  -  7  =  5 

13  -  8  =  5 

14  -  9  =  5 

12  -  6  =  6 

13  -  7  =  6 

14  -  8  =  6 

15  -  9  =  6 

13  -  6  =  7 

14  -  7  =  7 

15  -  8  =  7 

16  -  9  =  7 

14  -  6  =  8 

15  -  7  =  8 

16  -  8  =  8 

17  -  9  =  8 

15  -  6  =  9 

16  -  7  =  9 

17  -  8  =  9 

18  —  9  =  9 

65.  If  tlie  same  number  is  added  to  any  hvo  niim- 
bers,  the  difference  bettveen  the  resulting  numbers  is  the 
same  as  that  between  the  given  numbers.     Thus : 

The  diiFerence  between  7  and  2  is  5.  If  now  6  be 
added  to  both  7  and  2,  the  difference  between  the  re- 
sulting numbers,  13  and  8,  is  still  5. 

PROBI.KM    I. 

00.  To  subtract  one  number  from  another,  Avhen  each 
figure  of  the  subtrahend  is  equal  to,  or  less  than,  the  cor- 
responding figure  of  the  minuend,  counting  from  the  right, 

1.  Write  the  less  number  under  the  greater,  so  as  to 


SUBTRACTION. 


47 


place  the  figures  representing  the  corresponding  orders  of 
units  directly  under  each  other. 

2.  Begin  at  the  right,  and  subtract  each  figure  of  the 
subtrahend  from  the  figure  of  the  minuend  above  it;  the 
results  placed  under  the  figures  from  which  they  were 
obtained  Avill  express  the  difference  between  the  two 
numbers. 


(1.) 
From  5346 
Take   2145 


EXAMPLES. 

(2.)  (3.) 

7890  4567 

3450  1023 


(4.) 
67814 
6514 


Ans. 

3201                4440 

3544 

61300 

5. 

From  76503  take  65402. 

Ans. 

11101. 

6. 

From  84321  take  62100. 

Aiis. 

22221. 

7. 

From  54360  take  21030. 

Ans. 

33330. 

8. 

From  74215  take  3115. 

Alls. 

71100. 

9. 

From  21036  take  24. 

Ajis. 

21012. 

10, 

From  762137  take  1025. 

Ans.  ' 

761112. 

11. 

From  12345  take  2345. 

Ans. 

10000. 

12. 

From  54321  take  4321, 

Alls. 

50000. 

13. 

From  20037  take  10036. 

Ans. 

10001. 

(14.)               (15,) 

(16.) 

(17.) 

From 

45.13             78.64 

§3.105 

§41.043 

Take 

34.02              67.64 

2.104 

21.032 

Ans.  11.11  11.00 

18.  From  87.36  take  43.15. 

19.  From  96.125  take  5.01. 

20.  From  128.41  take  127.2 

21.  From  |45.16  t:ike  |3Lr 


§1.001 


§20.011 

Ans.  44.21 

A71S.  91.115. 

Ans.  1.21 

Ans.  §11.01 


48  SUBTIIACTION. 

22.  From  $3426.45  take  §113.23.      Ans.  3313.22. 

23.  From  §4327.871  take  §3216.461. 

Ans.  §1111.410. 

24.  From  §945.375  take  §334.275. 

Ans.  §611.100. 

25.  From  §12.032  take  §1.021.  Ans.  §11.011. 

26.  From  §119.457  take  §8.236.      Ans.  §111.221. 

PROBLiEM    II. 

G7.  To  find  the  difference  between  any  two  numbers 
whatever. 

EXAMPLES. 

1.   From  493  take  287. 

OPERATION. 
493 

287 

206  Ans. 

Here  7  can  not  be  taken  from  3,  because  7  is  larger 
than  3.  Mentally  add  10  units  to  the  3  units,  and  from 
the  sum  13  units,  take  7  units,  placing  the  difference, 
6  units,  under  the  figures  representing  units  of  the  first 
order. 

Since  now  the  minuend  has 'been  increased  by  10 
units,  we  must  increase  the  subtrahend  by  the  same 
number  of  units  to  preserve  the  true  difference. — (Vide 
65.) 

Mentally  add  1  unit  of  the  second  order  (which  is  the 
very  same  thing  as  10  units  of  the  first  order)  to  the  8 
units  of  the  second  order,  and  we  have  9  units  of  the 
second  order.     Proceed  now  as  in  66.     Hence, 


SUBTRACTION.  49 

Tt  U  L  E  . 

1.  Write  the  numhers  as  in  G6. 

2.  If  the  unit  figure  of  the  minuend  is  equal  to  or 
greater  than  the  unit  figure  of  the  subtrahend,  subtract  as 
in  (j6;  but  if  the  unit  figure  of  the  minuend  be  less  than 
that  of  the  subtrahetid,  mentally  add  10  ^o  the  upper 
figure  and  subtract  the  loiuer  figure  from  the  sum,  placing 
the  result  under  the  unit  column. 

3.  Add  1  to  the  next  figure  of  the  subtrahend,  and 
proceed  tvith  the  result  precisely  as  with  the  figures  repre- 
senting units  of  the  first  order. 

4.  Do  the  same  thing  to  the  figures  representing  each 

order  of  U7iits,  and  the  results  will  express  the  difference 

between  the  two  numbers. 

Remark  1, — The  work  may  be  verified  by  adding  the  remainder  to 
the  subtrahend. 

2.  From  7804  take  5936. 

OPERATION. 

7804  min. 
5936  sub. 


1868 


7804  proof. 

Remark  2. — The  pupil  should  say  at  once,  6  fi'om  14  leave  8;  4 
from  10  leave  6;  10  from  18  leave  8;  6  from  7  leave  1.  Do  not  say 
^from  4  you  can^t,  1  to  carry,  etc. 


(3.) 
From  5432 
Take  1685 

(4.) 
*  34.57 
23.19 

(5.) 
56.19 
24.396 

(6.) 
125.4 
37.236 

Alls.   3747 

11.38 
34.57 

31.794 

56.19 

88.164 

Proof  5432 
5 

125.4 

50 

SUBTRACTION. 

(7.)                          (8.) 

(9.) 

From 

$125,456                 $1243.18 

$7256.372 

Take 

87.25                      125.914 

199.20 

Ans. 

$38,206                 $1117.266 

$7057.172 

(10.) 

(11.) 

From  645.00037 

10000.0000 

Take         .00198 

.0111 

Ans.    644.99839 

9999.9889 

12. 

From  40000  take  9. 

Ans.  39991. 

13. 

From  123456789  take  87654321. 

Ans.  35802468. 

14. 

From  101010101  take  90909090. 

Ans.  10101011. 

15. 

From  303030303  take  40404040. 

A71S.  262626263. 

16. 

From  234702358  take  54321987. 

Ans.  180380371. 

17. 

From  1000000  take  1. 

Ans.  999999. 

18. 

From  1000000  take  .1. 

Ans.  999999.9. 

19. 

From  1000000  take  .01. 

Ans.  999999.99. 

20. 

From  34567  take  .003. 

Ans.  34566.997. 

21. 

From  1  take  .000001. 

Ans.  .999999. 

22. 

From  5  take  4.000001. 

Ans.  .999999. 

23. 

From  12.4  take  3.756. 

Ans.  8.644. 

24. 

From  100.25  take  75.12. 

Ans.  25.13. 

25. 

From  $100.25  take  $75.12. 

Ans.  $25.13. 

26. 

From  $20.05  take  $5.50. 

Ans.  $14.55. 

27. 

From  $90.00  take  $70,045. 

Ans.  $19,955. 

28. 

From  $1000  take  $1,111. 

Ans.  $998,889. 

SUBTRACTION.  51 

29.  From  6  dollars  take  5  mills.  Ans.  $5,995. 

30.  From  8  dollars  take  7  cents.  Ans.  $7.93. 

31.  America  was  discovered  in  A.  D.  1492  by  Chris- 
topher Columbus.  How  many  years  from  that  event  to 
A.  D.  1865  ?  A71S.  373  years. 

32.  George  Washington  was  born  in  A.  D.  1732,  and 
died  in  A.  D.  1799.     To  what  age  did  he  live? 

Ans.  67. 

33.  The  Declaration  of  Independence  was  published 
July  4,  1776.  How  many  years  have  intervened  up  to 
July  4,  1865?  A^is.  89. 

34.  Henry  Hudson  sailed  up  the  river  of  his  name, 
A.  D.  1609.  How  many  years  from  that  time  to  A.  D. 
1865?  Ans.2b6. 

35.  The  Mariners'  Compass  was  invented  in  A.  D. 
1302.     How  many  years  to  A.  D.  1865  ?     Aiis.  563. 

36.  What  length  of  time  from  the  birth  of  Francis 
Bacon,  A.  D.  1561,  to  A.  D.  1865?     Ans.  304  years. 

37.  What  length  of  time  from  the  birth  of  Shake- 
speare, A.  D.  1564,  to  A.  D.  1865  ?    Ans.  301  years. 

38.  What  length  of  time  from  the  birth  of  John 
Milton,  A.  D.  1608,  to  A.  D.  1865?    Ans.  257  years. 

39.  Pliny  died  in  A.  D.  17.  How  many  years  to  A. 
D.  1865  ?  A71S.  1848. 

40.  Sir  William  Herschel  was  born  in  A.  D.  1738,^ 
Galileo,  A.  D.  1564.     How  many  years  elapsed  from  the 
birth  of  the  one  to  that  of  the  other?  Ajis.  174. 

41.  Oliver  Cromwell  was  born  A.  D.  1599.  How 
many  years  from  that  time  to  the  death  of  Washington  ? 

A71S.  200. 

42.  Patrick  Henry  was  born  A.  D.  1 736.     How  many 


52  SUBTRACTION. 

years  from  that;  time  to  the  publication  of  the  Decla- 
ration of  Independence  ?  Aiis.  40. 

43.  The  Revolutionary  War  began  A.  D.  1775 ;  the 
last  war  with  Great  Britain,  A.  D.  1812.  How  many 
years  from  the  beginning  of  the  one  to  the  beginning 
of  the  other?  Ans.  37. 

44.  What  was  the  increase  in  the  population  of  New 
York  from  A.  D.  1850  to  A.  D.  I860?— (Vide  59,  Ex. 
33  and  34.)  Ans.  290104. 

45.  How  many  more  inhabitants  in  New  York  in 
1850  than  in  Philadelphia?  Ans.  175502. 

46.  How  many  more  in  1860  ?  A^is.  243122. 

47.  How  many  more  in  Boston  than  in  New  Orleans 
in  1850?  A71S.  20506. 

48.  How  many  more  in  New  Orleans  than  in  Cincin- 
nati? Ans.  939. 

49.  How  many  more  in  Cincinnati  than  in  Brooklyn  ? 

Ans.  18598. 

50.  How  many  more  in  Brooklyn  than  in  St.  Louis  ? 

Ans.  18978. 

51.  How  many  more  in  St.  Louis  than  in  Albany? 

A71S.  27097. 

52.  How  many  more  in  Albany  than  in  Pittsburg  ? 

A71S.  4162. 
53. .  How  many  more  in  New  York  than  in  Pittsburg  ? 

A71S.  468946. 

54.  The  polar  diameter  of  the  earth  is  7898.973 
miles;  the  equatorial  diameter,  7925.249  miles.  How 
much  greater  is  the  equatorial  than  the  polar  diameter  ? 

Ans.  26.276  miles. 

55.  The  length  of  a  degree  of  longitude  at  the  equator 


SUBTRACTION.  58 

is  69.161  miles ;  at  New  York  it  is  52.536  miles.     What 
is  the  difference  ?  A^is.  16.625  miles. 

56.  A  is  worth  §6542.37;  B  is  worth  $9341.95 ;  C 
is  worth  §18425.63.  How  much  are  all  three  together 
worth?  Ans.  §34309.95. 

How  much  is  B  worth  more  than  A  ? 

Ans.  §2799.58. 
How  much  more  is  C  worth  than  B? 

Ans.  §9083.68. 
How  much  more  is  C  worth  than  A? 

Ans.  §11883.26. 
How  much  more  is  C  worth  than  A  and  B  together? 

Ans.  §2541.31. 

57.  To  stock  a  farm,  the  land  of  which  was  worth 
§22475.96,  I  bought  two  horses  for  §327.80  ;  two  yoke 
of  oxen  at  §175.47  per  yoke  ;  five  cows  at  §27.36  each  ; 
a  pair  of  mules  for  §275;  and  sixty-seven  sheep  for 
§201.45.  How  much  more  is  the  land  worth  than  the 
stock?  Ans.  §21183.97. 

How  much  more  were  the  oxen  worth  than  the  horses  ? 

Ans.  §23.14. 
How  much  more  were  the  oxen  worth  than  the  cows  ? 

A71S.  §214.14. 
Which  were  worth  most,  the  oxen  and  horses  together, 
or  the  cows,  mules,  and  sheep  together,  and  by  how 
much?  Ans.  Oxen  and  horses,  by  §65.49. 

58.  A  merchant  bought  at  one  time  3476  yards  of 
cloth;  at  another,  5426  yards;  at  another,  4221  yards. 
He  sells  3210  yards  to  one  person,  and  4345  to  another. 
How  many  yards  has  he  left?  Ans.  5568  yards. 

59.  A  farmer  bought  of  a  merchant  broadcloth  to  the 


54  SUBTRACTION. 

value  of  $137.50;  cotton  cloth,  §93.45;  sugar,  |37.63; 
molasses,  114.37;  coffee,  §11.45;  flour,  §28.13.  He  pays 
the  merchant,  in  corn,  §123.65 ;  in  hay,  §47.24;  and  the 
balance  in  cash.     What  was  the  amount  of  cash  paid? 

Ans.  §151.64. 

60.  Bought  a  yoke  of  oxen  for  §150 ;  a  horse  for 
§237;  three  cows  for  §87.45;  and  sold  the  whole  for 
§500.     What  was  my  gain?  Ans.  §25.55. 

61.  The  length  of  a  pendulum  which  vibrates  once  a 
second,  at  London,  is  39.1393  inches.  One  ten-millionth 
of  the  meridian  distance  from  the  Equator  to  the  North 
Pole  is  39.37079  inches.     What  is  the  difference? 

Ans.  .23149  inches. 

62.  The  equatorial  circumference  of  the  earth  is 
24897.883  miles,  and  the  circumference  on  a  meridian 
is  24855.296  miles.     What  is  the  difference  ? 

Ans.  42.587  miles. 

63.  What  is  the  difference  between  25  dollars  1  cent 
4  mills  and  6  dollars  17  cents  9  mills  ? 

A71S.  §18.835. 

64.  What  is  the  difference  between  181  dollars  7 
cents  9  mills  and  140  dollars  9  cents  7  mills? 

A71S.  §40.982. 

65.  What  is  the  difference  between  9  dollars  5  cents 
3  mills  and  10  dollars  3  cents  5  mills? 


MULTIPLICATION.  65 


MULTIPLICATION. 


68.  Multiplication  is  the  operation  of  increasing 
one  number  as  many  times  as  there  are  units  in  another. 

69.  The  number  to  be  increased  is  called  the  multi- 
plicand. 

70.  The  number  indicating  how  many  times  the 
multiplicand  is  to  be  increased  is  called  the  multi- 
plier. 

Tl.    The  result  of  the  operation  is  called  the  product. 

SIGNS. 

72,  The  sign  X  is  called  sign  of  multiplication,  and 
when  placed  between  two  numbers,  signifies  that  they 
are  to  be  multiplied  together.     Thus,  the  expression, 

8x5-=40, 
is  read,  eight  multiplied   hy  five  equal  forty,  or   eight 
times  five  equal  forty,  or   eight  times  five  are  forty, 

73.  A  bar  or  imrenthesis  is  used  to  indicate  that 
several  numbers  are  to  be  taken  as  a  single  number,  thus : 


2+6X5--40,  or  (2-f  6)X5=-40;  but  2+6x5=32. 

•74.  When  two  or  more  numbers  are  multiplied  to- 
gether to  produce  a  single  number,  each  of  the  numbers 
involved  is  called  a  factor  of  the  product.  Thus,  in 
the  expression  3X2X5=30,  each  of  the  numbers,  3,  2, 
5,  is  a  factor  of  30. 

The  product  of  the  factors  is  a  composite  number. 


56 


MULTIPLICATION. 


75.  It  is  evident  that  3X5  is  the  same  as  5X3.     For 

3+3+3+3+3  is  the  same  as  5+5+5. 

76.  Any  number  multiplied  by  0  produces  0;  and 
any  number  multiplied  by  1  produces  the  number  itself. 

Thus,  8X0=0,  and  8X1=8. 


TABLE     OF     MULTIPLICATION. 


2x0  = 

0 

3x0 

=    0 

4x0 

=    0 

5x0=0 

2x1  = 

2 

3  X  1 

=    3 

4  X  1 

=    4 

5x1=5 

2x2  = 

4 

3x2 

=    6 

4x2 

=    8 

5  X  2  =  10 

2x3  = 

6 

3x3 

=    9 

4x3 

=  12 

5  X  3  =  15 

2x4  = 

8 

3x4 

=  12 

4x4 

=  16 

5  X  4  =  20 

2x5  = 

10 

3x5 

=  15 

4x5 

=  20 

5  X  5  =  25 

2x6  = 

12 

3x6 

=  18 

4  X  6 

=  24 

5  X  6  =  30 

2x7  = 

14 

3X7 

=  21 

4x7 

=  28 

5  X  7  =  35 

2x8  = 

16 

3x8 

=  24 

4x8 

=  32 

5  X  8  =  40 

2x9  = 

18 

3x9 

=  27 

4x9 

=  36 

5  X  9  =  45 

6x0  = 

0 

7x0 

=    0 

8x0 

=    0 

9x0=0 

6x1  = 

6 

7  X  1 

=    7 

8  X  1 

=    8 

9x1=9 

6x2  = 

12 

7x2 

=  14 

8x2 

=  16 

9  X  2  =  18 

6x3  = 

18 

7x3 

=  21 

8x3 

=  24 

9  X  3  =  27 

6x4  = 

24 

7x4 

=  28 

8x4 

=  32 

9  X  4  =  36 

6x5  = 

30 

7x5 

=  35 

8x5 

=  40 

9  X  5  =  45 

6  X  G  = 

36 

7x6 

=  42 

8x6 

=  48 

9  X  6  =  54 

6x7  = 

42 

7x7 

=  49 

8x7 

=  56 

9  X  7  =  63 

6x8  = 

48 

7x8 

=  56 

8x8 

=  04 

9  X  8  =  72 

6x9  = 

54 

7x9 

=  63 

8x9 

=  72 

9  X  9  =  81 

PROBIiEM    I. 

77.    To  multiply  any  number  by  any  other  number 
less  than  10. 

EXAINIPLES. 

1.   Multiply  357  by  7. 


MULTIPLICATIOX.  57 

OPERATION.  VERIFICATION. 

357  357 


49 

350 
2100 


7  357 

357 


357  vide  68  and  75. 

357 

357 


2490  357 


2499 


7  times  7  units  of  the  first  order  are  49  units  of  tlie 
first  order;  that  is,  9  units  of  i\iQ  first  order  and  4  of  the 
second. 

7  times  5  units  of  the  second  order  are  35'  units  of  the 
second  order;  that  is,  5  units  of  the  second  order,  and  3 
of  the  third. 

7  times  3  units  of  the  third  order  are  21  units  of  the 
tliird  order;  that  is,  1  unit  of  the  third  order  and  2  of 
the  fourth. 

The  sum  of  these  partial  products  is  the  product 
required. 

This  operation  may  be  contracted  thus : 

7  times  7  are  49.     Write  down  the  9  units  operation 
of  the  first  order  and  mentally  add  the  four      oerr 
units  of  the  second  order  to  the  35  units  of  7 

the  same  order,  making  39.      Write  down    

the  9  units  of  the  second  order,  and  men- 
tally add  the  3  units  of  the  third  order  to 
the  21  units  of  the  same  order,  making  24. 
Hence, 


58 


MULTIPLICATION. 


RULE, 


1.  Write  the  midtiplier  under  the  midtipliccmd,  so  that 
units  of  the  same  order  may  stand  under  each  other. 

2.  Ifultijyly  the  right-hand  figure  of  the  midtiplicand 
hy  the  multiplier^  and  set  doivn  the  figure  of  tlie  product 
representing  units  of  the  first  order  under  the  column  of 
units  of  that  order,  and  add  the  figure  representing  units 
of  the  second  order  to  the  product  of  the  second  figure  of 
the  multiplicand  hy  the  multii^lier.  Set  down  the  figure 
representing  units  of  the  second  order  under  the  units  of 
the  multiplicand  of  that  order,  and  add  the  figure  repre- 
senting units  of  the  third  order  to  the  next  product,  and  so 
on  till  all  the  figures  of  the  midtiplicand  have  been  mul- 
tij)lied.     The  result  is  the  product  required. 

(2.)  (3.)  (4.)  (5.) 

Multiply  1736  4530  7106  2400 

By  3  4  5  6 


Ans.        5208 

18120 

35530              14400 

Multiply  303479 
By                      2 

(7.) 
9854321 

:                        7 

(8.) 
123456789 
9 

Ans.        606958 

68980247 

1111111101 

9.   Multiply 

10.  Multiply 

11.  Multiply 

12.  Multiply 

13.  Multiply 

14.  Multiply 

456031  by  3. 
32467  by  8. 
10054  by  5. 
999999  by  9. 
5432  by  7. 
142857  by  7. 

Ans.  1368093. 

Ans.  259736. 

Ans.  50270. 

A71S.  8999991. 

Ans.  38024. 

Ans.  999999. 

MULTIPLICATION. 


59 


15.  Multiply  101010  by  9.  A71S.  909090. 

16.  Multiply  3421  by  4.  Ans.  13684. 

17.  Multiply  123456789  by  2,  3,  4,  5,  6,  7,  8,  9. 

18.  Multiply  987654321  by  2,  3,  4,  5,  6,  7,  8,  9.      ^ 

PROB1.EM11. 

78.    To  multiply  any  number  by  a  unit  of  any  order, 
that  is,  by  10,  100,  1000,  etc. 

RULE. 

An7iex  as  many  ciphers  to  the  right  of  the  multiplicand 
as  there  are  ciphers  in  the  multiplier. — (Vide  38,  II.) 

EXAMPLES. 


1. 

Multiply  357  by  1.— (Vide  76 

>') 

Ans.  357, 

2. 

Multiply  357  by  10. 

Am.  3570, 

3. 

Multiply  357  by  100. 

Ans.  35700, 

4. 

Multiply  358  by  1000. 

Ans.  358000, 

5. 

Multiply  3476  by  100. 

Ans.  347600, 

6. 

Multiply  35760  by  10. 

Ans.  357600, 

7. 

Multiply  473  by  lOOOOOO. 

A: 

ns.  473000000, 

8. 

Multiply  473000  by  1000. 

Ans.  473000000, 

9. 

Multiply  473000000  by  1. 

A: 

ns.  473000000, 

PROBIiEM     III. 

79.  To  multiply  any  number  by  a  figure  represent- 
ing units  of  any  order, 

(1.)  Consider  the  figure  as  representing  units  of  the 
first  order,  and  zvrite  the  numbers  as  in  77. 

(2.)  After  multiplying  annex  the  ciphers,  and  the  result 
will  he  the  j^roduct  required. 


60 


MULTIPLICATION. 


Multiply  357 
By  40 

Ans.       14280 

(5.) 

Multiply  4530 

By 


E  X  A  JI  P  L  E  S  . 

(2.)      (3.) 
357      1736 

200       70 


30 


71400    121520 
(6.) 
4530 
900 


(4.) 
1736 
400 

694400 

(7.) 
3456 
7000 


Ans.       135900 


24192000 
A71S.  6321000. 
A71S.  355300. 


4077000 

8.  Multiply  9030  by  700. 

9.  Multiply  7106  by  50. 

10.  Multiply  303479  by  20000. 

Ans.  6069580000. 

11.  Multiply  987654321  by  900000. 

Ans.   888888888900000. 

PROBIiEM    IV. 

80.   To  multiply  one  number  by  another. 


EXAMPLES. 

(1.)  (2.) 

Multiply  ...     357  Multiply     .     . 

By  .     .     ;     .     .    ^  By    ...     . 

Vide  77,  Ex.  3, 


4530 
934 


Vide  77,  Ex.  1,  2499 
Vide  79,  Ex.  1,  1428 
Vide  79,  Ex.  2,  714 


1812 

Vide  79,  Ex.  5,     1359 
Vide  79,, Ex.  6,  4077 


Ans. 


88179 


Ans. 


4231020 


Remark. — The  cipher  of  79,  Ex.  1,  need  not  appear  in  (he  opera- 
tion, and  so  of  the  ciphers  in  the  other  examples  referred  to.  Hence, 


I 


MULTIPLICATIOX. 


61 


1.  Write  iJie  niimhers  so  that  the  riglit-ltand  significant 
figures  of  the  multiplicand  and  multiplier  may  stand 
under  each  other. 

2.  Multiply  the  multiplicand  hy  each  figure  of  the  mul- 
tiplier, placing  the  first  figure  of  each  product  directly 
tinder  the  figure  used  in  multiplying. 

3.  Add  the  several  products  together  and  annex  to  their 
sum  all  the  ciphers  on  the  right  of  both  factors.  The 
result  is  the  product  required. 

Remark  1. — The  second  point  of  the  rule  does  not  apply  to  the 
ciphers  on  the  right  of  either  factor. 


Multiply 

By     . 


Ans. 


(3.) 


(4.) 


1476 
470 

Multiply  . 
By      .    . 

.     .     34300 
.     .     4310 

10332 
5904 

343 

1029 
1372 

fip.q790 

Ans.      .     .     147833000 

Remark  2. — When  ciphers  occur  between  the  significant  figures 
of  the  multiplier,  their  product  into  the  multiplicand  need  not  ap- 
pear in  the  operation. 


(5.) 


(6.) 


Multiply 
By.    . 


.     459 
.     307 

3213 
1377 


Multiply 
By     .    , 


2134 
5004 


8536 
10670 


Ans 140913         Ans. 

7.  Multiply  46834  by  4060. 

8.  Multiply  47042  by  47042. 


.  .  .  10678536 
Ans.  190146040. 
Ans.  2212949764. 


62 


MULTIPLICATION. 


9.  Multiply  123  by  125. 

10.  Multiply  328  by  67. 

11.  Multiply  75432  by  47. 

12.  Multiply  678954  by  24. 

13.  Multiply  789563  by  570. 

14.  Multiply  1579126  by  1710. 

15.  Multiply  67853  by  8765. 

16.  Multiply  3678543  by  4567. 

17.  Multiply  492  by  625. 

18.  Multiply  1312  by  335. 

19.  Multiply  603456  by  94. 

20.  Multiply  1357908  by  144. 

21.  Multiply  2368689  by  190. 

22.  Multiply  8432  by  6350. 

23.  Multiply  27496  by  1658. 

24.  Multiply  82488  by  555. 

81.  When  one  or  both  factors 


Ans.  15375. 

Ans.  21976. 

Ans.  3545304. 

Ans.  16294896. 

Ans.  450050910. 

^Tis.  2700305460. 

Ans.  594731545. 

Ans.  16799905881. 

A71S.  307500. 

Ans.  439520. 

Ans.  56724864. 

Ans.  195538752. 

Ans.  450050910. 

Ans.  53543200. 

A71S.  45588368. 

Ans.  45780840. 

are  decimals. 


EXAMPLES. 


(1-) 


Multiply 357 

By      .....       .     24.7 

2499 
1428 
714 


Ans 8817.9 

In  this  example,  since  the  mul- 
tiplier is  decreased  ten  times, 
(vide  38,  II,)  the  product  ought 
to  be  decreased  ten  times,  which 
is  done  by  placing  one  figure 
to  the  right  of  the  decimal 
point. 


(2.) 


Multiply 85.7 

By 24.7 

2499 
1428 
714 


Ans 881.79 

In  this  example,  since  the  mul- 
tijDlicand  is  also  decreased  ten 
times,  (vide  80,  Ex.  1,)  the  pro- 
duct ought  to  be  decreased  ten 
times  more  than  in  Ex.  1,  which 
is  done  by  putting  the  decimal 
point  one  place  further  to  the  left 


MULTIPLICATION. 


63 


Hence : 

(1.)  Proceed  'precisely  as  in  80. 

(2.)  Place  the  decimal  ijoint  in  the  lyroduct  so  as  to  cause 
as  many  figures  to  stand  on  its  right  as  there  are  figures 
on  the  right  of  the  point  in  both  the  factors,  supplying 
any  deficiency  by  prefixing  ciphers. 

(3.)  (4.)  (5.) 

Multiply       45.9  4.59  .459 

By  3.07  3.07  .307 


3213                         3213 

3213 

1377                        1377 

1377 

Ans.       140.913                   14.0913 

.140913 

(6.)                      (7.) 

(8.) 

Multiply         .0008                  .00008 

.000008 

By                  .0007                      .007 

.07 

Ans.        .00000056             .00000056 

.00000056 

(9.) 

(10.) 

Multiply         .0716 

.1234 

By                  1.326 

1234 

Ans.         .0949416                  152.2756 

11.  Multiply  $3.57  by  7. 

Ans.  $24.99. 

12.  Multiply  ^3.57  by  40. 

Ans.  $142.80. 

13.  Multiply  $3.57  by  200. 

Ans.  $714.00. 

14.  Multiply  ?3.57  by  .7 

Ans.  $2,499. 

15.  Multiply  171.61  by  365.           Ans.  $26137.65. 

16.  Multiply  $0.93  by  63. 

Ans.  $58.59. 

17.  Multiply  $13.75  by  43. 

Ans.  $591.25. 

18.  Multiply  $4.68  by  169. 

Ans.  $790.92. 

64 


MULTIPLICATION. 


19.  Mult 

20.  Mult 

21.  Mult 

22.  Mult: 

23.  Mult 

24.  Mult 

25.  Mult 

26.  Mult 

27.  Mult 

28.  Mult 

29.  Mult 

30.  Mult 

31.  Mult 

32.  Mult 


ply 
ply 

ply 

ply 
ply 
ply 
ply 
ply 
ply 
ply 
ply 
ply 
ply 


,^0.057  by  84C). 
1132.55  by  369"! 
^0.299  by  69. 
§69.748  by  144. 
§3.75  by  47. 
$6.79  by  163. 
§1.375  by  19. 
§4.57  by  18. 
§15.89  by  9. 
§0.75  by  125. 
§58.90  by  45. 
§0.058  by  .07 
§0.904  by  .025 
§1.287  by  .9 


Ans,  §48.222. 

Ans.  §48910.95. 

Ans.  §20.631. 

Ans.  §10043.712. 

Ans.  §176.25. 

Ans.  §1106.77. 

Ans.  §26.125. 

Ans.  §82.26 

Ans.  §143.01. 

A71S.  §93.75. 

Ans.  §2650.50. 

Ans.  §0.00406. 

Ans.  §0.022600. 

Ans.  §1.1583. 


82.  In  Multiplication  the  j-^'^'oduct  is  always  of  the 
same  name  as  the  multiplicand^  and  the  multiplier  in  the 
operation  must  be  considered  as  simply  a  number  without 
name.     Thus:  §75X47=-§3525.— (Vide  166.) 


PKACTICAL     APPLICATION. 

1.  What  will  47  oxen  cost,  at  §75  each?  ' 

Ans.  $3525. 

2.  If  a  man  walk  23  miles  a  day,  how  far  will  he 
walk  in  17  days?  Ans.  391  miles. 

3.  If  a  vessel  sail  451  miles  a  day,  how  far  will  it 
sail  in  9  days  ?  Ans.  4059  miles. 

4.  What  will  495  yards  of  cloth  cost,  at  11  dollars  a 
yard?— (Vide  75.)  Am.  §5445. 

5.  What  will  569  hogsheads  of  molasses  cost  at  $37 
each?  Ans.  §21053. 


MULTir  LIGATION.  '65 

6.  What  will  451  bales  of  cotton  cost  at  |53  per  bale? 

Alls.  §23903. 

7.  There  arrived  in  market  18  wagons  during  one 
■week,  each  wagon  containing  6  bales  of  cotton,  worth 
|56  per  bale.     How  much  was  the  cotton  worth  ? 

Ans.  §6048. 

8.  Bought  8  bales  of  cotton,  each  bale  containing 
530  pounds,  worth  §0.13  per  pound.  How  much  did  I 
pay  for  it?  .  A^is.  §551.20. 

9.  I  purchased  at  one  time  3  bales  of  cotton,  each 
bale  weighing  554  pounds,  at  §0.11  per  pound;  at 
another  time,  5  bales,  each  weighing  535  pounds,  at 
§0.12  per  pound.     What  was  the  whole  worth  ? 

Ans.  §503.82. 

10.  I  sold  the  cotton  in  the  preceding  example  at 
§0.115  per  pound.     Did  I  make  or  lose,  and  how  much? 

Ans.  lost  §5.065. 
,  11.   I  bought  27  hogsheads  of  molasses,  each  contain- 
ing 63  gallons,  at  §0.53  per  gallon.     How^  much  did  1 
pay  for  it?  Ans.  §901.53. 

12.  I  sold  a  sack  of  hops,  weighing  396  pounds,  at 
§0.113  per  pound.     How  much  did  I  get  for  it? 

Ans.  §44.748. 

13.  What  cost  5342  barrels  of  flour  at  §8.50  per 
barrel  ?  A71S.  §45407. 

14.  One  chest  of  tea  contains  69  pounds,  and  costs 
§0.299  per  pound;  another  chest  contains  74  pounds, 
and  costs  §0.274  per  pound.  How  much  is  lost  by  sell- 
ing the  whole  at  §0.28  per  pound?  Ans.  §0.867 

15.  What  cost  169  boxes  of  oranges  at  §6.71  per 
box?  Ans.  §1133.99. 


66  MULTIPLICATIOX. 

16.  What  cost  357  barrels  of  potatoes  at  §2.47  per 
barrel?  Ans.  ^SS1.19. 

17.  I  purchase  243  casks  of  butter,  each  containing 
57  pounds,  at  §0.34  per  pound,  and  sell  the  same  at 
S0.40  per  pound.     How  much  do  I  gain  ? 

Ans,  §831.06. 

18.  My  vineyard  produces  5342  bottles  of  wine,  at  a 
cost  of  §0.37  per  bottle,  and  I  retail  the  wine  at  §1.25 
per  bottle.     How  much  do  I  clear?      Ans.  §4700.96. 

19.  Two  passenger  trains  of  cars  meet  at  a  station, 
and  each  train  runs,  on  an  average,  37  miles  an  hour. 
Suppose  them  to  start  at  the  same  time,  how  far  apart 
will  they  be  at  the  end  of  17  hours  ? 

Ans.  1258  miles. 

20.  A  passenger  train  and  a  freight  train  of  cars 
start  from  a  given  station  and  run  in  the  same  direction. 
The  passenger  train  moves  at  the  rate  of  37  miles  an 
hour,  and  the  freight  train  19  miles  an  hour.  How  far 
apart  will  they  be  at  the  end  of  13  hours? 

A71S.  234  miles. 
How  far  apart  had  they  run  in  different  directions? 

Ans.  728  miles. 

21.  A  drover  bought  357  oxen,  at  the  rate  of  §47  a 
head.  In  going  to  market,  12  oxen  fell  through  a  bridge 
and  were  killed.  The  cost  of  driving  the  remainder  to 
market  was  §5  a  head.  The  oxen  were  then  sold  at  an 
average  of  §49  each.     What  was  the  loss  ? 

A71S.  §1599. 

22.  How  much  would  have  been  gained  by  selling  the 
oxen  in  the  preceding  example  at  §55  a  head? 

Ans.Un. 


MULTIPLICATION. 


67 


MERCHANTS'   BILLS. 

83.   A  Bill  is  a  written  statement  of  articles  bought 
or  sold,  their  prices,  entire  cost,  etc. 

Buffalo,  April  26, 1865. 
Messrs.  J.  H.  Reed  &  Co.,  Chicago,  111., 

Bought  of  D.  Ransom  &  Co., 
Terms  cash.  121  Main  Street. 


10 

5 

Gross  Trask's  Magnetic  Ointmeut,  @  $21 

"      Ransom's  Hive  Syrup  Tolu,  @  $24 

.  $210 
.    120 
.    160 
.    160 
.      95 

00 

00 

5 

5 

"      Mrs.  Winslow's  Soothing  Syrup,  @  $32 

"      Brown's  Bronchial  Troches,  @  $32 

00 
00 

5 

"      Judson's  Mt.  Herb  Pills,  @  $19 

00 

$745 

00 

Received  payment.               , 

D.  Ransom  &  Co., 
By  John  M.  Sabin,  Cl'k. 
10  Cases  per  Lake  Shore  Railroad. 

(2.) 
New  York,  Jan.  1, 1865. 
Mr.  James  Bryant,  Mobile,  Ala., 

Bought  of  A.  Smith  &  Co. 

24 

Pounds  of  Tea,  @  $0.63 

$15jl2 

55 

Barrels  of  Potatoes,  @  $5 

275  00 

98 

Pounds  of  Raising  @  $0.45 

41 

807 

85 

85 

Barrels  of  Shad,  @  $9.50 

50 

^1139 

47 

6'p^i-'i 


^ 


68  MULTIPLICATION. 

(8.) 

New  York,  Jan.  1,  1865. 
Mr.  Lewis  Rollings, 

•     To  Otis  Howe  &  Co.,  Dr. 

1864. 

Sept.  13.     To  30  doz.  Wool  Hose @  $1.50 

"     13.      "     6  prs.  Gloves  @  $0.75 

"     13.       "    4  doz.  Napkins @  $1.20 

Oct.   12.      "     4    "     Shirt  Bosoms @  $2.40 

"      12.       "  22  yds.  Drilling : .@  $0.10 

"     12.      "     6    "     Broadciotli @  $4.00 


$90.10 
Received  payment. 

S.  Billings, 
For  Otis  Howe  &  Co. 

-    •  (4.) 

^       St.  Louis,  July  15,  1865 
Mr.  S.  H.  DeCamp. 

To  A.  T.  Campbell,  Dr. 

1865. 

Jan.     5.  To  12    doz.  Scythes @  $10.00 

"        9.  "  6J   "     Hoes @    $7.00 

Feb.     1.  "  10      "     Rakes @    $1.80 

March  4.  "  3    Plows @  $11.00 

"      4.  "  7    doz.  Pitchforks @    $9.50 

"      1.  "  9      «     Padlocks @  $24.00 

«    12.  "  1    Coffee-mill @    $5.00 

July  15.     Settled  by  due  bill $504.00 

A.  T.  Campbell. 


DIVISION.  69 


DIVISION. 


84.  Division  is  the  operation  of  finding  how  many 
times  one  number  is  contained  in  another. 

85.  The  number  to  be  divided  is  called  the  dividend. 
80.    The  member  by  which  to  divide  is  called  the  di- 
visor. 

8T.  The  number  of  times  the  divisor  is  contained  in 
the  dividend  is  called  the  quotient. 

88.  IVlien  there  is  a  number  left  after  dividing,  it  is 
called  the  remainder. 

SIGN. 

89.  The  sign  -^-  is  known  as  the  sigri  of  division,  and 
when  placed  between  two  numbers,  it  signifies  that  the 
former  is  to  be  divided  by  the  latter.  Thus,  the  ex- 
pression, 

40-f-5=8, 

is  read,  forty  divided  by  five  equal  eight,  or  five  in  forty 
eight  times. 

OO.  Operations  in  division  are  carried  on  under  two 
forms.     Thus : 

(1.)  (2.)  (3.) 

5)13  5)13(2  13 1 5 

-:.  o  10  10  T 

2-3  —  — 

3  3 


70 


DIVISION. 


The  first  is  known  as  the  method  of  Short  Division. 

The  second  is  known  as  the  method  of  Long  Di- 
vision. 

In  either  of  these  forms,  13  is  the  dividend,  5  is  the 
divisor,  2  is  the  quotient,  and  3  the  remainder. 

In  the  operations  of  Long  Division,  the  divisor  is  some- 
times written  as  in  (3.) 

Ol.  The  remainder  added  to  the  product  of  iJie  divisor 
and  quotient  produces  the  dividend.     Thus : 

5x2+3=13. 

(1.)  Any  number  divided  by  itself  produces  1 ;  and 
(2.)  Any  number  divided  by  1  produces  the  number 
itself.     Thus,  7^7=1,  and  7-^1=7. 


TABLE  OF 

division. 

2-4-2=1 

3  -f-3  =  1 

4  -r-4=  1 

5-4-5  =  1 

4-^2  =  2 

6  -r-3  =  2 

8-^4  =  2 

10  -^  5  =  2 

6h-2  =  3 

9^3  =  3 

12  -T-4  =  3 

15  -=-  5  =  3 

8-^2  =  4 

12  -H  3  =  4 

16  --4  =  4 

20  H-  5  =  4 

10  -i-  2  =  5 

15  -V-  3  =  5 

20  ^  4  =  5 

25  -4-  5  =  5 

12  -V-  2  =  6 

18  -V-  3  ==  6 

24  -H  4  =  6 

30  -4-  5  =  6 

14  ~  2  =  7 

21  -4-  3  =  7 

28  ~  4  =  7 

35  -4-  5  =  7 

16  -^  2  =  8 

24  -^  3  =  8 

32  -T-  4  =  8 

40  -4-  5  =  8 

18  -r-  2  =  9 

27  -r-  3  =  9 

36  ~  4  =  9 

45  -4-  5  =  9 

6  -=-6  =  1 

7  -T-7  =  1 

8-4-8  =  1 

9-4-9  =  1 

12  --  6  =  2 

14  -^  7  =  2 

16  ~  8  =  2 

18  -4-  9  =  2 

18  -^  6  =  3 

21  -4-  7  =  3 

24  H-  8  =  3 

27  -T-  9  =  3 

24  ^  6  =  4 

28  -^  7  =  4 

32  -4-  8  =  4 

36  -4-  9  =  4 

30  ^  6  =  5 

35  -i-  7  =  5 

40  -4-  8  =  5 

45  -4-  9  =  5 

36  -T-  6  =  G 

42  -v-  7  =  6 

48  -T-  8  =  6 

54  -4-  9  =  6 

42  ~  6  =  7 

49  -V-  7  =  7 

56  -4-  8  =  7 

63  -4-  9  =  7 

48  -f-  6  =  8 

56  -^  7  =  8 

64  -^  8  =  8 

72  -4-  9  =  8 

54  -r-  6  =  9 

63  -^  7  =  9 

72  -:-  8  =  9 

81  -^  9  =  9 

DIVISION.  71 

SHORT    DIVISION. 
02.    To  divide  a  number  by  a  number  less  than  10. 


EXAMPLES. 

(1.) 

(2.) 

(3.) 

7)2499 

5)35530 

3)3698 

357  7106  1232-2 

7  is  contained  in  24  units  of  the  tldrd  order,  3  units 
of  the  same  order  and  3  over.  Write  down  the  3  units 
of  the  third  order,  and  prefix  the  3  over  to  the  9  units 
of  the  second  order,  making  39. 

7  is  contained  in  39  units  of  the  second  order  5  units 
of  the  same  order  and  4  over.  Write  down  the  5  units 
of  the  second  order,  and  prefix  the  4  over  to  the  9  units 
of  the  first  order,  making  49. 

7  is  contained  in  49  units  of  the  first  order ,  7  units  of 
the  same  order.  Then  357  is  the  quotient. — (Vide  77, 
Ex.  1.) 

In  the  second  example,  we  may  then  say  at  once,  5  in 
35  give  7;  5  in  5  give  1 ;  5  in  3  give  0;  5  in  30  give  6. 
—(Vide  77,  Ex.  4.) 

In  the  third  example,  3  in  3  give  1 ;  3  in  6  give  2 ;  3 
in  9  give  3 ;  3  in  8  give  2,  and  2  remainder.     Hence-, 

B  u  L  E  . 

1.  Write  the  divisor  on  the  left  of  the  dividend,  as  in 
90,(1.) 

2.  Find  hoiv  many  times  the  divisor  is  contained  in 
the  left-hand  figure,  or,  if  that  is  smaller  than  the  divisor^ 


72 


DIVISION. 


in  two  of  the  left-hand  figures  of  the  dividend,  mid  ivrite 
the  quotient  diredhj  under  the  figure  of  the  lowest  order 
used. 

3.  Prefix  the  remainder  to  the  figure  of  the  next  lower 
order,  and  divide  as  before,  continuing  the  tvorJc  till  all 
the  figures  have  been  used.  The  result  is  the  quotient  re- 
quired. 

Remark  1. — The  final  remainder,  if  tliere  is  any,  is  to  be  written 
as  in  90,  (1.) 

(4.)  (5.)  (6.) 

2)606958     7)68980247     9)1111111101 


303479              Vide  77,  Ex.  7. 

Vide  77,  Ex. 

7. 

Divide  1368093  by  3. 

Ex.9. 

.8. 

Divide  259736  by  8. 

Ex.  10. 

9. 

Divide  8999991  by  9. 

Ex.  12. 

10. 

Divide  5208  by  3. 

Ex.2. 

11. 

Divide  38024  by  7. 

Ex.  13. 

12. 

Divide  999999  by  7. 

Ex.  14. 

13. 

Divide  13684  by  4. 

Ex.  16. 

14. 

Divide  340974  by  9.  • 

Ans.  37886. 

15. 

Divide  10101  by  3. 

Ans.  3367. 

16. 

Divide  270192  by  6. 

'Ans.  45032. 

17. 

Divide  3530380  by  5. 

Ans.  706076. 

18. 

Divide  4236456  by  6. 

Ans.  706076. 

19. 

Divide  4942532  by  6. 

Ans. 

20. 

Divide  818181  by  9. 

Ans.  90909. 

21. 

Divide  103701  by  3. 

Ans.  34567. 

22. 

Divide  2751840  by  1,  2,  3,  4, 

etc. 

23. 

Divide  $4725  by  4. 

Jlem.  §1. 

24. 

Divide  13257  by  8. 

Pern.  |1. 

DIVISION. 

25. 

Divide  $4321  by  2. 

Rem.  $1. 

26. 

Divide  16721  by  3. 

Rem.  $1. 

27. 

Divide  $3454  by  5. 

Rem.  $4. 

28. 

Divide  $2348  by  7. 

Rem.  $3. 

29. 

Divide  4532  by  9. 

Rem.  5. 

30. 

Divide  17623  by  3. 

Rem.  1. 

31. 

Divide  30407  by  5. 

Rem.  2. 

32. 

Divide  40321  by  2. 

Rem.  1. 

83. 

Divide  76541  by  9. 

'Rem.  5. 

34. 

Divide  $451  by  5. 

Rem.  $1. 

73 


*  Remark  2. — The  divisor  may  be  written  under  the  remainder 
with  a  line  between  them,  the  whole  being  considered  as  forming 
a  part  of  the  quotient.     Thus,  (vide  134,) 

(35.)  (36.)  (37.) 

2)7  3)16  8)245 

3i  5i  81f 

The  answer  of  (35)   is  read  three  and  one  half. — 
(Vide  8.) 

The  answer  of  (36)  is  read  five  and  one  third. 

The  answer  of  (37)  is  read  eighty-one  and  two  thirds. 


88. 

Divide  4725  by  4. 

Ans.  1181  J. 

39. 

Divide  8890  by  4. 

Ans.  972f . 

40. 

Divide  5341  by  5. 

Ans.  1068J. 

41. 

Divide  8459  by  5. 

Ans.  691f. 

42. 

Divide  3005  by  6. 

Ans.  5001. 

43. 

Divide  $451  by  2. 

Ans.  $225J. 

44. 

Divide  $650  by  3. 

Ans.  $216f . 

45. 

Divide  $121  by  4. 

Ans.  $301. 

46. 

Divide  $154  by  4. 

Ans.  $38f . 

47. 

Divide  $3459  by  7. 

7 

Ans.  $4944. 

74  DIVISION. 

48.  Divide  P258  by  8.  ^^s.  *407f . 

49.  Divide  $1111  by  9.  Ans.  |123|. 

50.  Divide  $2222  by  5.  Ans.  §444f . 

51.  Divide  $3333  by  6.  Ans.  $555|. 

52.  Divide  $4444  by  7.  Ans.  $634f . 

53.  Divide  $5005  by  8.  Ans.  $625|. 

54.  Divide  $3003  by  9.  Ans.  $333f . 
55.-  Divide  $221  by  2.  Ans.  $110|. 

Remark  8;— r-Ciphers  may  be  annexed  to  tlie  dividend,  and  the 
division  continued  till  there  is  no  remainder,  or  it  may  terminate 
at  any  convenient  point.  The  figures  of  the  quotient,  after  annexing 
ciphers,  are  decimals.     Thus, 

(56.)  (57.)  (58.) 

4)375.00  3)$5678.000  9)$4573.000 

93.75  $1892.6661  $508,111^ 

The  answer  of  (57)  is  1892  dollars  66  cents  6  mills 
and  two-thirds  of  a  mill. 

59.  Divide  4725  by  4.  ^ns.  1181.25. 

60.  Divide  3890  by  4.  Ans.  972.5. 

61.  Divide  5341  by  5.  ^ws.  1068.2. 

62.  Divide  3459  by  5.  Ans.  691.8. 

63.  Divide  3005  by  6.  Ans.  500.83|. 

64.  Divide  $451  by  2.  Ans.  $225.50. 

65.  Divide  $4357  by  6.  ^m.  $726.166|. 

66.  Divide  $1  by  2,  3,  4,  5,  6,  7,  8,  9. 

Ans.  $0.50,  $0,331,  $0.25,  $0.20,  $0.166|,  $0.142f, 

$0,125,  $0.111f 

03.'  When   the   divisor,    dividend,  or  both   contain 
decimals, 

1.   If  (he  numher  of  decimal  figures  in  the  divisor 


DIVISION.  75 

exceeds  that  in  the  dividend^  make  it  equal  in  both   hy 
annexing  ciphers  as  decimals  to  the  dividend. 

2.  Divide  as  in  92,  and  place  the  decimal  point  in  the 
quotient,  so  as  to  cause  as  many  figures  to  stand  on  its 
right  as  all  the  decimal  figures  used  in  the  dividend  exceed 
those  in  the  divisor,  supplying  any  deficiency  of  quotient 
figures  hy  prefixing  ciphers. — (Vide  81.) 

EXAMPLES. 

Divide  249.9  by  7. 

OPERATION. 

7)249.9 

35.7 

Here  the  dividend  has  one  decimal  figure,  the  divisor 
none. 

2.  Divide  2499  by  .7. 

OPERATION. 

.7)2499.0     (Vide  93, 1.) 
3570 

Here  the  dividend  has  one  decimal  figure,  and  the 
divisor  one.     The  quotient  is  an  integral  number. 

3.  Divide  24.99  by  .07. 

OPERATION. 

.07)24.99 

357 
Here   the    dividend   has   two    decimal   figures,   and 


76  DIVISION. 

the  divisor  two,  and  the  quotient  is  then  an  integral 
number. 

4.  Divide  249.9  by  .0007. 

OPERATION. 

.0007)249.9000     (Vide  93,  1.) 

357000 
Here  the  dividend  has  four  decimal  figures,  and"^  .3 
divisor  four.  „ 

5.  Divide  .2499  by  7. 

OPERATION. 

7).2499 

.0357  (Vide  93,  2,  last  clause.) 

Here  the  dividend  has  four  decimal  figures,  the  divisor 
none.  One  cipher  had  to  be  prefixed  to  the  three  quo- 
tient figures. 

6.  Divide  52.08  by  .03. 


OPERATION. 

.03)52.08 

1736  Ans. 
.03 

52.08  Proof 

'.     (Vide  91.) 

7. 

8. 

9. 
10. 
11. 

Divide  375  by  A 

Divide  430  by  .008 
Divide  4725  by  .4 
Divide  32570  by  .08 
Divide  43.21  by  .002 

Ans.  937.5 

Ans.  53750, 

Ans.  11812.5 

Ans.  407125. 

Ans.  21605. 

DIVISION.  77 

12.  Divide  7.621  by  .0003.  Am.  25403.331. 

13.  Divide  .00214  by  .002.  Am.  1.07, 

14.  Divide  .03017  by  .007.  Am.  4.31. 

15.  Divide  93.276  by  .007.  Am.  13325.142f . 

16.  Divide  48.35  by  .005.  Am.  9670. 

17.  Divide  14535  by  9,  .9,  .09,  .009,  .0009,  .00009, 
and  .000009.  Am.  1615,  16150,  161500,  etc. 

B4:.   To  divide  an  integral  number  or  decimal  by  a 
UMt  of  any  order,  that  is,  by  10,  100,  1000,  etc., 

T  Remove  the  decimal  point  of  the  dividend  as  many 
figures  to  the  left  as  are  indicated  hy  the  ciphers  in  the 
divisor.— {VidiQ  38,  II.) 

EXAMPLES. 

1.  Divide  3570  by  10,  (vide  78,  Ex.  2.) 

Am.  357. 

2.  Divide  35700  by  100.  Am.  357. 

3.  Divide  357  by  10.  Am.  35.7. 

4.  Divide  357  by  100.  Am.  3.57. 

5.  Divide  35.7  by  10.  Am.  3.57. 

6.  Divide  31.4  by  100.  Am.  .314. 

7.  Divide  §451.30  by  10.  Am.  $45.13. 

8.  Divide  §4321  by  100.  Am.  §43.21. 

9.  Divide  §23456  by  1000.  Am.  §23.456. 

10.  Divide   §1.00   among   10   men,  100  men,  1000 
men.  Am.  §0.10,  §0.01,  §0.001. 

11.  Divide  §45.20  equally  among  10  men;  100  men; 
1000  men.  -Aws.  §4.52;  §0.452;  §0.0452. 

12.  Divide  §3120  equally  among  10000  men. 

A71S.  §0.312. 


78 


DIVISION. 


LONa    DIVISION. 

05.    To  divide  a  number  by   any   number  greater 
than  10. 


EXAMPLES. 

1.   Divide  1716  by  11. 

(1.)  We  find  how  many  times 
11  is  contained  in  17,  and  write 
the  quotient  figure  1  to  the  right 
of  the  dividend. 

(2.)  Multiply  the  divisor  11 
by  the  quotient  figure  1,  and 
place  the  product  11  under  17. 

(3.)    Subtract  11  from  17,  and 


OPERATION. 

11)1716(156 

11 

61 
55 

66 


Am. 


place  the  remainder  6  under  11. 

(4.)    Bring    down     the     next  q 

figure  1  of  the  dividend  to  the 

right  of  the  6,  and  proceed  with  the  result  61  exactly  as 
with  17.     That  is, 

(1.)    11  in  61  are  5,  which  is  placed  in  the  quotient. 

(2.)    Then  5  times  11  are  55,  which  is  placed  under 
61.     - 

(3.)    55  from  61  are  6. 

(4.)    Bring  down  the  next  figure  6  of  the  dividend. 
In  the  resulting  number  66  the  divisor  11  is  contained 

Hence, 


6  times,  with  no  remainder. 


RULE, 


(1.)   Find  how  many  times  the  whole  divisor  is  con- 
tained in  the  same  niuiiher  of  figures  on  (he  left  of  the 


DIVISION.  79 

dividend  considered  as  representing  so  many  units ;  or, 
if  this  number  is  smaller  than  the  divisor,  find  how 
many  times  the  divisor  is  contained  in  a  number  of  figures 
of  the  dividend^  greater  by  one  than  the  number  of  figures 
in  itself,  and  place  the  quotient  on  the  right  of  the  divi- 
dend. 

(2.)  Multiply  the  whole  divisor  by  this  quotient  figure, 
and  place  the  p)roduct  under  the  figures  of  the  dividend 
just  compared  ivith  the  divisor. 

(3.)    Subtract  the  product  from  the  figures  above  it. 

(4.)  To  the  right  of  the  difference  bring  doion  the  next 
figure  of  the  dividend. 

(5.)  If  the  new  dividend  is  noiv  smaller  titan  the  divi- 
sor, place  a  cipher  in  the  quotient,  and  bring  down  the  next 
figure  of  the  dividend;  but  if  the  dividend  is  equal  to  or 
larger  than  the  divisor,  proceed  in  obtaining  the  second 
figure  of  the  quotient  as  ivith  the  first,  and  so  on  till  all 
the  figures  are  brought  down. 

Remark  1. — Notice  that  no  product  can  be  greater  than  the 
figures  of  the  dividend  above  it. 

Remark  2. — Notice  that  no  difference  can  be  equal  to  or  greater 
than  the  divisor. 

2.  Divide  1716  by  12.  Ans.  143. 

3.  Divide  1716  by  13.  Ans.  132. 

4.  Divide  3360  by  14.  Ajis.  240. 

5.  Divide  3360  by  15.  Ans.  224. 

6.  Divide  3360  by  16.  Ans.  210. 

7.  Divide  7429  by  17.  Ans.  4S7. 

8.  Divide  7429  by  19.  Ans.  391. 

9.  Divide  7429  by  23.  Ans.  323. 
10.  Divide  26691  by  21.  Ans.  1271. 


80  DIVISION. 

11.   Divide  7371  bj  91.  Arts.  SI. 

(12.)  (13.) 

91)7371(81  184)56488(307 

728  552 

91  1288     (Vide  Rule,  5.) 

91  1288 


0  0 

14.  Divide  56488  by  92.  Ajis.  614. 

15.  Divide  84487  by  97.  Ans.  871. 

16.  Divide  24108  by  98.  Ans,  246. 

17.  Divide  24108  by  246.  Ans.  98. 

18.  Divide  5768  by  56.  A^is.  103. 

19.  Divide  48071  by  53.  .      Ans.  907. 

20.  Divide  2448  by  16,  17,  and  18. 

Ans.  153,  144,  and  136. 

21.  Divide  6072  by  22,  23,  and  24. 

Ans.  276,  264,  and  253. 

22.  Divide  5544  by  36,  44,  and  56. 

Ans.  154, 126,  and  99. 

23.  Divide  12259  by  13,  23,  and  41. 

Ans.  943,  533,  and  299. 

(24.)  (25.) 

509)461663(907  999)708291(709 

4581  6993 


3563  8991 

3563  8991 


26.  Divide  73647  by  147  and  167. 

Ans.  501  and  441. 


DIVISION.  81 

27.  Divide  36942  by  131  and  141. 

Ans.  282  and  262. 

28.  Divide  526587  by  191  and  919. 

Ans.  2757  and  573. 

29.  Divide  3203055  by  711  and  901. 

Ans,  4505  and  3555. 

30.  Divide  193581  by  137  and  157. 

Ans.  1413  and  1233. 

31.  Divide  404278  by  431  and  134 

Ans.  938  and  3017. 

32.  Divide  670592745  by  54321. 

OPERATION. 

670592745154321 
54321    12345 


127382 

108642  (Yide  90,  (3.) 

187407 
162963 


244444 

217284 

271605 
271605 

33.  Divide  45780840  by  82488.      Ans.  555- 

34.  Divide  1234549380  by  12345.  Ans.  100004. 

35.  Divide  497477808387  by  987. 

Ans.  504030201. 

36.  Divide  49419533647761876  by  9876. 

Ans.  5004003002001. 


82  DIVISION. 

Remark  3. — If  there  are  cipliers  on  the  right  of  the  divisor, 
they  may  be  neglected  in  the  operation,  together  with  a  like  num- 
ber of  figures  on  the  right  of  the  dividend. 


37.   Divide  34705  by 

700. 

38.  Divide  34705  by 

760. 

OPERATIONS 

3. 

(37.) 

(38.) 

7'00)347'05 

76^0)3470'5(45 

49—405 

304 

430 
380 

505 

The  remainder  of  (37)  is  405,  and  that  of  (38)  is  505. 

39.  Divide  3521  by  200.  Rem,  121. 

40.  Divide  3521  by  30.  Bern.  11. 

41.  Divide  4561  by  40.  Rem.  1. 

42.  Divide  3457  by  50.  Rem.  7. 

43.  Divide  7654  by  70.  Rem  24. 

44.  Divide  3420  by  90.  Arts.  38. 

45.  Divide  1716000  by  12000.  Ans.  143. 

46.  Divide  1716000  by  130000.  .        Rem.  26000. 

47.  Divide  3360000  by  1400000.       Rem.  560000. 

48.  Divide  73647000  by  1470000.     Rem.  147000. 

49.  Divide  123456789  by  2000000. 

Rem.  1456789. 

Remark  4. — In  decimals  observe  the  Rule  in  93  and  92,  Rem.  3. 

50.  Divide  143  by  25. 

51.  Divide  143  by  125. 


DIVISION.       1 

(50.) 
25)143.00(5.72 
125 

OPERATIONS. 

(51.) 
125)143.000(1.144 
125 

180 
175 

50 
50 

180 
125 

550 

500 

0 

500 
500 

83 


52.  Divide  206.166492  by  4.123. 

53.  Divide  .102048  by  3189. 

OPERATIONS. 

(52)  (53) 

206.166492|  4.123  .102048|3189 

206.15  50.004  9567  .000032 


16492  (Yide  90,  (3.)       6378 

16492                6378 

54. 

Divide  4567  by  25,  125,  and  625. 

- 

Ans.  182.  68,  etc. 

55. 

Divide  1460  by  16,  32,  and  64. 

Ans.  91.  25,  etc. 

56. 

Divide  7623  by  40,  80,  and  1600. 

Ans.  190.  575,  etc. 

57. 

Divide  143  by  15. 

58. 

Divide  100  by  24. 

84  I  DIVISION. 


'  OPERATIONS. 

(57.)  (58.) 

15)143.00(9.53  X  24)100.00(4.16|| 

135  \  96 


80     (Vide  92,  Rem.  2.)  40 

75  24 

50  ,  160 

45  ^  144 

5  16 

59.  Divide  140.913  by  3.07  and  45.9. 

60.  Divide  281.826  hj  30.7  and  4.59. 

Ans.  9.18  and  61.4. 

61.  Divi^  .0949416  by  1.326.  Ans.  .0716. 

62.  Divide  152.^756  by  .1234.  Ans.  1234. 

63.  Divide  $14|.80  equally  among  40  men. 

Ans.  13.57  each. 

64.  Divide  §714.00  equally  among  200  men. 

Ans.  13.57  each. 

65.  Divide  §2.499  equally  among  7  men. 

Ans.  §0.357  each. 

66.  Divide  $26137.65  equally  among  365  men. 

Ans.  $71.61  each. 

67.  Divide  $58.59  equally  among  63  men. 

Ans.  $0.93  each. 

68.  Divide  $10043.712  equally  among  144  men. 

Ans.  $69,748  each. 

69.  Divide  $176.25  equally  among  47  men. 

Ans.  $3.75  each. 


DIVISION.  85 

70.  Divide  $48910.95  equally  among  369  men. 

Ans.  $132.55  each. 

71.  Divide  |20.631  equally  among  69  men. 

Ans.  fO.299  each. 
96.    In  Division  the  quotient  is  always  of  the  same 
name  as  the  dividend;  and  the  divisor  in  the  operation 
must  be  considered  as  simply  a  number  without  name. 
Thus,  $3525^47-?75.— (Vide  167.) 

PRACTICAL   APPLICATIONS. 

1.  If  47  oxen  cost  $3525,  what  will  one  ox  cost? 

Ans.  $75. 

2.  If  a  man  can  walk  391  miles  in  17  days,  how  far 
can  he  walk  in  one  day  ?  Ans.  23  miles. 

3.  A  vessel  has  sailed  4059  miles  in  9  days.  What 
was  the  average  rate  per  day?  Ans.  451  miles. 

4.  I  sell  cloth  to  the  amount  of  $5445,  at  the  rate  of 
$11  per  yard.     How  many  yards  are  sold  ? 

Ans.  495  yards. 

5.  Eleven  men  divide  5445  boards  equally  among 
them.     How  many  does  each  have  ? 

Ans.  495  boards. 

6.  If  5445  marbles  are  equally  divided  among  495 
boys,  how  many  does  each  have  ?       Ans.  11  marbles. 

7.  If  495  yards  of  cloth  are  sold  for  $5445,  what  is 
the  average  price  per  yard?  Ans.  $11. 

8.  If  569  hogsheads  of  molasses  cost  $21053,  what  is 
the  price  per  hogshead  ?  Ans.  $37. 

9.  I  buy  molasses  to  the  amount  of  $21053,  at  the 
rate  of  $37  per  hogshead.  How  many  hogsheads  are 
bought  ? 


86  DIVISION. 

10.  If  $21053  are  equally  divided  among  37  men, 
how  many  dollars  does  each  man  receive? 

11.  There  arrived  in  market  a  number  of  wagons, 
each  loaded  with  12  bales  of  cotton,  worth  |56  per  bale. 
The  cotton  was  sold  for  $12096.  How  many  wagons 
came  to  market? 

12.  A  train  of  18  cars  brought  down  the  Mobile  and 
Ohio  Railroad  cotton  to  the  amount  of  $12096,  at  the 
rate  of  $56  per  bale.  How  many  bales  did  each  car 
carry,  supposing  the  cotton  equally  distributed? 

13.  Eighteen  cars  took  into  Memphis  $12096  worth 
of  cotton,  each  car  being  loaded  with  12  bales.  What 
was  the  cotton  valued  at  per  bale? 

14.  The  exact  distance  round  the  earth  at  the  equa- 
tor, is  24897.883  miles,  which  is  divided  into  360 
degrees.     What  is  the  length  of  each  degree  ? 

Ans.  69.161  miles. 

15.  How  long  would  it  take  a  man  to  travel  round 
the  earth  at  the  equator,  traveling  at  the  rate  of  60 
miles  per  day  ?  A718.  414.9647  J  days. 

16.  Since  the  earth  turns  on  its  axis  once  in  24  hours, 
how  far  does  any  point  on  the  equator  move  per  hour  ? 
Per  minute?     Per  second? 

Last  Ans.  .2881699  mHeS. 

17.  When  the  moon  is  240,000  miles  from  the  earth, 
how  long  would  it  take  a  car  to  make  the  distance,  at 
the  rate  of  45  miles  per  hour  ?      Ans.  5333 Jf  hours. 

18.  When  the  sun  is  95,000,000  miles  from  the  earth, 
how  long  would  a  car  be  occupied  in  making  the  dis- 
tance, running  at  the  rate  of  57  miles  per  hour  ? 

Ans.  16666661  f  hours. 


PROPERTIES    OP   INTEGRAL   NUMBERS.  87 


t 


PROPERTIES  OF  INTEGRAL  NUMBERS, 


DEFINITIONS. 

97.  Any  number  is  exactly  divisible  by  another  when 
the  q^tient  is  an  integral  number.  Thus,  51  is  exactly 
divisible  by  17,  because  the  quotient  3  is  an  integral 
number. — (Vide  7.)  But  51  is  not  exactly  divisible  by 
25,  because  the  quotient,  2j^^  contains  a  fractional 
number. — (Yide  8.) 

98.  An  EVEN  NUMBER  is  one  that  is  exactly  divisible 
by  2.     The  even  numbers,  then,  are  2,  4,  6,  8,  10,  etc. 

99.  An  ODD  NUMBER  is  one  that  is  not  exactly  divisible 
by  2.     The  odd  numbers  are,  then,  1,  3,  5,  7,  9,  11,  etc. 

100.  A  PRIME  NUMBER  is  One  that  is  exactly  divisible 
by  no  number  except  itself  and  1. 

Remark  1. — No  even  number^  except  2,  can  he  a  prime  number. 

Remark  2. — The  prime  numbers  less  than  100,  are 

1,     2,     3,     5,     7,  11,  13,  17,  19,  23,  29,  31,  37, 

41,  43,  47,  53,  59,  61,  67,  71,  73,  79,  83,  89,  97. 

101.  A  COMPOSITE  NUMBER  is  the  product  of  two  or 
more  prime  numbers. — (Vide  74.)     Thus, 

24=2X2X3X2. 

Remark. — Any  composite  number  is  exactly  divisible  by  either  of  its 
prime  factors,  or  by  the  product  of  any  two  or  more  of  them.  Thus, 
30  is  divisible  by  2,  3,  or  5,  or  by  2X3=::6,  2X5=10,  3X5=15. 


88  PROPERTIES    OP   INTEGRAL    NUMBERS. 

10!S.  Any  two  numbers  are  said  to  be  prime  with 
respect  to  each  other  when  no  factor  of  the  one  is  a  factor 
of  the  other.  Thus,  26=2X13,  and  63-=3x3x7,  are 
prime  with  respect  to  each  other. 

103.  A  COMMON  DIVISOR  of  two  or  more  numbers  is 
a  number  which  exactly  divides  each  of  them.  Thus,  2  is 
a  common  divisor  of  12  and  20. 

Remark. — Numbers  which  are  prime  with  respect  to  each  pther,  have 
no  common  divisor  other  than  1. 

104.  The  GREATEST  COMMON  DIVISOR  of  two  or  more 
numbers  is  the  greatest  number  which  exactly  divides 
each  of  them.  Thus,  4  is  the  greatest  common  divisor  of 
12  and  20. 

105.  A  MULTIPLE  of  a  number  is  the  product  of  that 
number  by  some  other  number.  Thus,  the  multiples  of 
2  are  4,  6,  8,  10,  etc.  The  multiples  of  5  are  10,  15, 
20,  etc.     The  multiples  of  36  are  72,  108,  144,  etc. 

106.  A  COMMON  MULTIPLE  of  two  or  more  numbers 
is  a  number  that  is  exactly  divisible  by  each  of  them. 
Thus,  84  is  a  common  multiple  of  2,  3,  6,  and  7,  because 
it  is  exactly  divisible  by  each  of  them. 

107.  The  LEAST  COMMON  MULTIPLE,  of  two  or  more 
numbers  is  the  least  number  that  is  exactly  divisible  by 
them.  Thus,  84  is  the  least  common  multiple  of  7,  12, 
and  14,  because  it  is  the  least  number  exactly  divisible 
by  each  of  them.  ^ 

Remark. — Every  number  is  the  least  common  multiple  of  all  its  prime 
factors,  or  of  all  its  divisors.  Thus,  42  is  the  least  common  multiple 
of  2,  3,  and  7,  its  prime  factors,  or  of  2,  3,  6,  7,  14,  and  21,  all  its 
divisors,  except  1  and  42,  which  may  be  included. 


plloperties  of  integral  numbers.  89 

Properties  of  the  Integral  Numbers,  from  2  to  13, 
inclusive. 

108.  Every  number,  and  no  other,  is  exactly  divisible 
by  2,  wJiose  rigid-hand  figure  is  a  0,  or  an  even  number. 
Thus, 

120,  374,  336,  95678,  are  exactly  divisible  by  2. 

341,  753,  255,  24683,  are  not  exactly  divisible  by  2. 

101]>..  Every  number,  and  no  other,  is  exactly  divisible 
by  3,  the  sum  of  whose  digits  is  divisible  by  3.     Thus, 

1035,  1305,  1350,  5031,  are  exactly  divisible  by  3, 
because  1+0+3+5=9  is  exactly  divisible  by  3. 

liO.  Every  number,  arid  no  other,  is  exactly  divisible 
by  4,  tvhose  two  right-hand  figures,  considered  as  repre- 
senting a  number  less  than  100,  are  exactly  divisible 
by  4.     Thus, 

120,  324,  428336,  are  exactly  divisible  by  4,  because 

20,  24,  36,  are  exactly  divisible  by  4.     But 

121,  463,  782354,  are  not  exactly  divisible  by  4,  because 

21,  63,  54,  are  not  exactly  divisible  by  4. 

111.  Every  number,  and  no  other,  is  exactly  divisible 
by  5,  whose  right-hand  figure  is  a  0,  or  a  S.     Thus, 

3240,  4535,  etc.,  are  exactly  divisible  by  5. 

112.  Every  numher,  and  no  other,  is  exactly  divisible 
by  6,  which  is  even,  and  exactly  divisible  by  3.  Thus, 
534,  2136,  6348,  etc.,  are  exactly  divisible  by  6. 

113.  Every  number,  and  no  other,  is  exactly  divisi- 
ble by  7,  when  the  difference,  of  the  sum  of  the  products 
produced  by  multiplying  the  figures  standing  in  the  odd 
PERIODS,  by  1,  3,  2,  respectively,  counting  from  right  to 


90  PROPERTIES    OF   INTEGRAL   NUMBERS. 

leftj  and  the  sum  of  the  products  of  the  even  periods, 
found  in  the  same  way,  is  exactly  divisible  by  7.     Thus, 

6,  883,  905, 
is  exactly  divisible  by  7,  because  the  difference  between 
the  sum  of  5+18+6=29=5Xl+9x2+6Xl=products 
in  odd  periods;  and  of  3+24+16=43=3x14-8X3+ 
8x2=products  in  even  periods,  which  is  14,  is  exactly 
divisible  by  7. 

Remark  1. — In  practice,  the  multiplication  and  addition  should 
be  done  mentally,  and  as  soon  as  the  sum  amounts  to  7  or  more 
omit  the  7,  and  use  the  surplus.     Thus, 

5  and  4  are  2  and  6  are  1. 
3  and  3  are  6  and  2  are  1. 
Remark  2. — If  the  figure  7  or  a  0  is  found  in  the  number  it  need 
not  be  multiplied.     Thus,  34,  707,  205. 

5  and  4  are  2  and  4  are  6  and  2  are  1; 
and  the  number  is  not  exactly  divisible  by  7,  but  will  have  a  re- 
mainder of  1. 

114.  Every  number,  and  no  other,  is  exactly  divisi- 
ble by  8,  whose  three  right-hand  figures,  considered  as 
representing  a  number  less  than  1000,  are  exactly  di- 
visible by  8.     Thus, 

387528;  135752,  etc.,  are  exactly  divisible  by  8,  because 
528;        752,  are  exactly  divisible  by  8. 

115.  Every  number,  and  no  other,  is  exactly  divisi- 
ble by  9,  the  sum  of  whose  digits  is  divisible  by  9. 
Thus, 

14454,  44451,  54414,  etc.,  are  exactly  divisible  by  9, 
because  4+4+4+1+5=18,  is  exactly  divisible  by  9. 

IIG.  Every  number,  and  no  other,  is  exactly  divisi- 
ble by  11,  when  the  difference  of  the  sujn  of  the  figures 


PROPERTIES    OF    INTEGRAL    NUMBERS.  91 

standing  in  the  odd  places^  and  the  sum  of  the  figures 
standing  in  the  even  places,  counting  either  way,  is 
exactly  divisible  by  11.  Thus,  64532146723  is  exactly 
divisible  by  11,  because  the  difference  between  27=6+5 
+2+4+7+3,  sum  in  odd  places,  and  16=4+3+1+6 
+2,  sum  in  even  places,  which  is  11,  is  exactly  divisible 
by  11. 

117.  Every  number,  and  no  other,  is  exactly  divisible 
by  12,  which  is  exactly  divisible  by  3  and  4. 

118.  The  proposition  in  reference  to  7  is  true  also  of 
13,  if  the  numbers  1,  10,  9,  are  substituted  respectively 
for  1,  .3,  2. 

Remark  1. — Any  number  composed  of  two  full  periods^  with  the  same 
figures  in  each,  and  the  same  arrangement,  is  exactly  divisible  by  7  or  13. 

144144,  305305,  101101,  352352,  111111,  are  eacli  exactly  divisi- 
ble by  7  or  13. 

Remark  2. — The  number  1001,  or  any  multiple  of  it,  is  exactly  di- 
visible by  7,  11,  or  13. 

FACTORING. 

119.  To  resolve  a  number  into  its  prime  factors, 

1.  Divide  the  number  by  any  prime  number  that  will 
exactly  divide  it. 

2.  Divide  the  quotient  by  any  prime  number  that  will 
exactly  divide  it. 

3.  Continue  thus  till  a  quotient  of  1  is  obtained. 
The  several  divisors  are  the  prime  factors  sought, 

EXAMPLES. 

1.   What  are  the  prime  factors  of  13860  . 


92 


PROPERTIES    OF   INTEGRAL   NUMBERS. 


Vide  108. 

Vide  109. 
Vide  111. 
Vide  113. 
Vide  116. 


OPERATION. 

2 13860 

2    6930 
,  3    3465 

5    1155 

7 
11 


231 


33 


The  prime  factors  of 
13860  are  2,  2,  3,  3, 
5,  7,  11. 


2.  What  are  the  prime  factors  of  4,  6,  S,  12,  14,  and 
16?  A)is.  2  and  2,  etc. 

3.  What  are  the  prime  factors  of  18,  20,  22,  26,  28, 
and  30  ?  A71S.  2,  3  and  3,  etc. 

4.  What  are  the  prime  factors  of  32,  34,  36,  38,  40, 
and  42?  Ans.  2,2,  2,  2,  and  2. 

5.  What  are  the  prime  factors  of  44,  45,  46,  48,  49, 
and  50  ?  Ans.  2,  2,  and  11,  etc. 

6.  What  are  the  prime  factors  of  all  the  composite 
numbers  between  50  and  100  ?— (Vide  100,  Rem.  2.) 

A71S.  51=3X17,  52=2X2X13,  etc. 

7.  What  are  the  prime  factors  of  121, 144,  169,  196, 
225,  256,  289,  324,  361,  400,  441,  and  484. 

Ans.  121=11X11,  etc. 

8.  What  are  the  prime  factors  of  143,  187,  209,  253, 
319,  341,  451,  561,  737,  913,  and  957. 

Ans.  143=11X13,  etc. 


PROPERTIES    OF   INTEGRAL   NUMBERS.  93 

9.  What  are  the  prime  factors  of  371,  413,  469,  497, 
623,  1001,  3003,  1309,  1463,  and  1771? 

Last  Ans.  7,  11   and  23. 

10.  What  are  the  prime  factors  of  2940,  4620,  5460, 
7140,  7690,  930,  1330,  1610,  6020,  and  4350? 

Last  Ans.  2,  3,5,  5   and  29. 

11.  Wliat  are  the  prime  factors  of  744  ? 

Ans.  2,  2,  2,  3  and  31. 

12.  What  are  the  prime  factors  of  1680  ? 

Ans.  2,  2,  2,  2,  3,  5  and  7. 

13.  What  are  the  prime  factors  of  636  ? 

Ans.  2,  2,  3  and  53. 

14.  What  are  the  prime  factors  of  1080? 

A71S.  2,  2,  2,  3,  3,  3  and  5. 

15.  What  are  the  prime  factors  of  5000? 

Ans.  2,  2,  2,  5,  5,  5,  and  5. 

16.  What  are  the  prime  factors  of  221,  299,  and  387? 

Ans.  13  and  17,  etc. 

17.  What  are  the  prime  factors  of  2431?         A^is. 

GREATEST    COMMON    DIVISOR. 

ISO.  To  find  the  greatest  common  divisor  of  two  or 
more  numbers, 

Resolve  the  numbers  into  their  prime  factors,  and  then 
find  the  2?roduct  of  such  as  are  common  to  all  the  num- 
bers. The  result  will  be  the  greatest  common  divisor 
sought. 

EXAMPLES. 

1.  What  is  the  greatest  common  divisor  of  168,  420, 
and  6006? 


94 


PROPERTIES    OF   INTEGRAL   NUMBERS. 


OPERATION 

168 

2 

420 

84 

2 

210 

42 

3 

105 

21 

5 

35 

7 

7 

7 

1 

1 

2 

6006 

3 

3003 

7 

1001 

11 

143 

13 

13 

The  factors  common  to  all  the  numbers  are  2,  3,  and 
7.  Hence,  2X3X7  =  42,  is  their  greatest  common 
divisor. 

2.  What  is  the  greatest  common  divisor  of  2,  4,  and 
6  ?  Ans.  2. 

3.  What  is  the  greatest  common  divisor  of  4,  6,  and 
8?     '  Ans.  2. 

4.  What  is  the  greatest  common  divisor  of  6,  8,  and 
10  ?  Ans.  2. 

5.  What  is  the  greatest  common  divisor  of  2,  4,  and 
8?  ^7is.  2. 

6.  What  is  the  greatest  common  divisor  of  3,  6,  and 
9?  Ans.  3. 

7.  What  is  the  greatest  common  divisor  of  4,  8,  and 
12?  Ans.  4. 

8.  What  is  the  greatest  common  divisor  of  8,  12, 
and  20?  A7is.4.     . 

9.  What  is  the  greatest  common  divisor  of  12,  18, 
and  24?  Am.  6. 


L 


PROPERTIES    OF   INTEGRAL   TSTUMBERS.  95 

10.  What  is  the  greatest  common  divisor  of  24,  36, 
and  48  ?  Ans.  12. 

11.  AVhat  is  the  greatest  common  divisor  of  14,  21, 
and  35  ?  Ans.  7. 

12.  What  is  the  greatest  common  divisor  of  26,  39, 
and  52?  Aiis.  IS. 

13.  What  is  the  greatest  common  divisor  of  16,  24, 
and  48?  ^  Ans.S. 

14.  What  is  the  greatest  common  divisor  of  252, 180, 
and  288  ?  Ans.  36. 

•    15.    What  is  the  greatest    common   divisor   of  144, 
196,  and  256  ?  Ans.  4. 

16.  What  is  the  greatest  common  divisor  of  744, 
1680,  636,  and  1080  ?— (Vide  119,  Examples  11,  12, 
13,  and  14.)  Ans.  2X2X3=12. 

17.  What  is  the  greatest  common  divisor  of  375, 
1100,  120,  and  1440?  Ans.  5. 

18.  What  is  the  greatest  common  divisor  of  780, 
1560,  720,  and  1008?  Ans.  12. 

19.  What  is  the  greatest  common  divisor  of  144, 
196,  256,  and  324?  Ans.  4. 

20.  What  is  the  greatest  ^common  divisor  of  143, 
187,  209,  and  253?  Ans.  11. 

21.  W^hat  is  the  greatest  common  divisor  of  216, 
408,  740,  and  810  ?  Ans.  2. 

22.  What  is  the  greatest  common  divisor  of  187, 
209,  52,  and  161  ?— (Vide  103,  Rem.)  Ans.  1 . 

121.  The  greatest  common  divisor  of  two  numbers  is 
the  greatest  common  divisor  of  the  less  of  the  two  num- 
bers, and  the  difference  found  by  taking  the  largest 
multiple  of  the  less  number,  which  is  smaller  than  the 


96  PROPERTIES    OF    INTEGRAL    NUMBERS. 

greater,  from  the  greater  number.     Thus,  the  greatest 
common  divisor  of 

36  and  136, 
is  also  the  greatest  common  divisor  of 
36  and  136— 108-=28, 
108  being  3X36,  the  largest  multiple  of  36  less  than 
136.     (Vide  105.)     This  must  be  so.     For  any  divisor 
of  36  also  divides  36x3,  and  since  it  divides  36X3,  if 
it  divides  36x3+28,  it  must  divide    28.     There   can, 
then,  be  no  common  divisor  of  36,  and  36x3+28,  -which 
is  not  a  common  divisor  of  36  and  28.     Therefore  the 
greatest  common  divisor  of  36  and  28  is  the  greatest 
common  divisor  of  36  and  136.     Hence, 

\'22.  To  find  the  greatest  common  divisor  of  two 
numbers, 

1.  Write  the  numbers  on  a  line  far  enough  from  each 
other  to  draw  one  or  two  perpendicular  lines  between 
them. 

2.  Find  the  greatest  multiple  of  the  smaller,  that  is  less 
than  the  larger,  and  subtract  it  from  the  larger  number. 
,i.--.3.    Find  the  greatest  multiple  of  the   remainder,  less 
than  the  smaller  number,  and  subtract  it  from  the  smaller 
number. 

4.  Find  the  greatest  multiple  of  this  remainder,  less 
than  the  previous  remainder,  and  subtract  as  before,  and 
continue  the  work  till  there  is  no  remainder. 

The  last  integral  remainder  will  be  the  greatest  common 
divisor  sought. 

EXAMPLES. 

1.  What  is  the  greatest  common  divisor  of  36  and 
136  ? 


PROPERTIES    OF    INTEGRAL    NUMBERS. 


97 


OPERATIONS. 


(1.) 


(2.) 


36 

28 

8 
8 


136 

108 

28 
24 


36 

28 

8 
8 


136 

108 

28 
24 


0   4  Am.  0  4  Ans. 

Remark. — The  multiples  are  found  precisely  as  in  division,  and 
the  quotient  figures  may  be  retained  as  in  (1),  or  neglected  as  in  (2). 

2.  What  is  the  greatest  common  divisor  of  285  and 
465  ?  Ans.  15. 

3.  What  is  the  greatest  common  divisor  of  532  and 
1274  ?  Ans.  14. 

4.  What  is  the  greatest  common  divisor  of  693  and 
1815  ?  Ans.  33. 

OPERATIONS. 


285 

465 

532$ 

21274 

693 

1815 

180 

285 

420$ 

U064 

429 

1386 

105 

180 

112] 

L  210 

264 

429 

75 

105 

981 

L  112 

165 

264 

30 

75 

14^ 

r    98 

99 

165 

30 

60 

98 

66 

99 

0 

15 

0 

33 

66 

66 

0 

5.   What  is  the  greatest  common  divisor  of  1054  and 
1426?  Ans.  62. 

9 


98  PROPERTIES    OF   INTEGRAL   NUMBERS. 

6.  What  is  the  greatest  common  divisor  of  725  and 
3175?  Ans,2^. 

7.  What  is  the  greatest  common  divisor  of  4585  and 
6685  ?  Am.  35. 

8.  What  is  the  greatest  common  divisor  of  4605  and 
5505?  Ans.U. 

9.  What  is  the  greatest  common  divisor  of  636  and 
1080?  of  744  and  1680?  of  972  and  1260?  of  3471 
and  1869  ?  Ans.  12,  24,  36,  and  267. 

10.  What  is  the  greatest  common  divisor  of  1137  and 
9475?  of  3447  and  9575?  of  2359  and  8425?  of  1903 
and  4325  ?  Ans.  379,  383,  337,  and  173. 

11.  What  is  the  greatest  common  divisor  of  117869 
and  137773?  ^ns.  311. 

Remark, — If  there  are  inore  than  two  numbers,  ,find  the  greatest 
common  divisor  of  any  two  of  them,  and  then  of  this  divisor  and  a  third 
number,  and  so  on  till  all  the  numbers  have  been  used.  The  last  integral 
remainder  will  be  the  greatest  common  divisor  of  all  the  members. 

12.  What  is  the  greatest  common  divisor  of  285, 465, 
and  35?— (Vide  Ex.  2.)  Ans.  5. 

13.  What  is  the  greatest  common  divisor  of  532, 
1274,  and  21?— (Vide  Ex.  3.)  Ans.  7. 

14.  What  is  the  greatest  common  divisor  of  1815, 
693,  66,  and  88?— (Vide  Ex.  4.)  Ans.  11. 

15.  What  is  the  greatest  common  divisor  of  620, 
1116,  and  1488  ?  Ans.  124. 

16.  What  is  the  greatest  common  divisor  of  1054, 
1426,  and  2263?  Ans.  31. 

17.  What  is  the  greatest  common  divisor  of  233,  587, 
and  653?— (Vide  103,  Rem.)  Ans.  1. 

18.  What  is  the  greatest  common  divisor  of  739, 
503,  and  439?  *  Ans.  1, 


I 


E^ 


PROPERTIES'  OF   INTEGRAL   NUMBERS.  99 

19.  What  is  the  greatest  common  divisor  of  97343, 
139639,  and  206193?  Ans.  311. 

LEAST   COMMON   MULTIPLE. 

123.  To  find  the  least  common  multiple  of  two  or 
more  numbers, 

1.  Write  the  numbers  in  a  line,  with  convenient  in- 
tervals. 

2.  Divide  by  any  prime  number  that  ivill  exactly  divide 
any  two  or  more  of  them,  and  write  the  quotients  and 
numbers  not  exactly  divisible  in  a  line  below  the  given 
numbers. 

3.  Divide  this  line  by  any  prime  number  as  before,  and 
continue  to  divide  till  no  prime  number,  except  1,  will 
exactly  divide  any  two  numbers  in  the  line. 

4.  Multiply  the  divisors  and  numbers  left  in  the  last 
line  together. 

The  product  will  be  the  least  common  multiple 
sought. 

Remark, — If  any  of  the  given  numbers  will  exactly  divide  any  other 
of  them,  it  may  be  neglected  in  the  operation,  and  not  affect  the  result. 

EXAMPLES. 

1.  What  is  the  least  common  multiple  of  4,  8,  9, 
and  21. 

OPERATION. 

8,  9,  21 


8,  3,     7 
Hence,  3X8x3X7=504,  is  the  least  common  mul- 
tiple required.  ' 


100 


I'KOPEIITIES    OF   INTEGRAL   NUMBERS. 


The  figure  4  is  neglected  in  the  operationj  because  it 
will  exactly  divide  8. 

2.  What  is  the  least  common  multiple  of  8,  64,  and 
72?  ^^s.  576. 

3.  What  is  the  least  common  multiple  of  26,  39,  and 
52  ?  A71S.  156. 

4.  What  is  the  least  common  multiple  of  14,  56,  and 


196? 


(2.) 


2 

64, 

72 

2 

32, 

36 

2 

16, 

18 

8, 

9 

OPE 

13 

RATIONS. 

(3.) 
39,  52 

3,     4 

A71S.  392. 

(4.) 
2 

2 


56, 

196 

28, 

98 

14, 

49 

2, 

7 

(2.)       • 
2X2X2X8X9=576. 


(3.) 
13X3X4=156. 


(4.) 

2X2X2X7X7=392. 

5.   What  is  the  least  common  multiple  of  2,  3,  4,  5, 
6,7,8,9,12? 

OPERATION. 


2 

5,  7,  8,  9,  12 

2 

5,  7,  4,  9,    6 

8 

5,  7,  2,  9,    3 

X 

5,  7,  2,  3,    1 
6X7X2X3=2520  Ans 

PROPERTIES    OF   INTEGRAL   NUMBERS.  101 

6.  What  is  the  least  common  multiple  of  7,  8,  and 
14?  of  5,  25,  and  50?  of  2,  4,  and  6? 

Ans.  56,  50,  and  12. 

7.  What  is  the  least  common  multiple  of  4,  6,  and  8  ? 
of  6,  8,  and  10  ?  of  2,  4,  and  8  ?      Ans.  24, 120,  and  8. 

8.  What  is  the  least  common  multiple  of  3,  6,  and  9  ? 
of  4,  8,  and  12  ?  of  8,  12,  and  20  ? 

Ans.  18,  24,  and  120. 

9.  What  is  the  least  common  multiple  of  12,  18,  and 
24  ?  of  24,  36,  and  48  ?  of  8,  32,  and  16  ? 

Ans.  72,  144,  and  32. 

10.  What  is 'the  least  common  multiple  of  8,  18,  and 
36  ?  of  21,  42,  and  14  ?  of  7,  14,  and  70  ? 

Ans.  72,  42,  and  70. 

11.  What  is  the  least  common  multiple  of  9,  18,  and 
27  ?  of  12,  16,  and  20  ?  of  5,  10,  and  15  ? 

Ans.  54,  240,  and  30. 

12.  What  is  the  least  common  multiple  of  2,  6,  and 
8  ?  of  7,  14,  and  21  ?  of  3,  4,  and  5  ? 

Ans.  24,  42,  and  60  ? 

13.  What  is  the  least  common  multiple  of  2,  3,  and 
4  ?  of  2,  5,  and  7  ?  of  3,  7,  and  11. 

_  Ans.  12,  70,  and  231. 

14.  What  is  the  least  common  multiple  of  14,  28, 
and  98?  of  8,  14,  and  35  ?  of  24,  25  and  32? 

Ans.  196,  280,  and  2400. 

15.  What  is  the  least  common  multiple  of  63, 12,  84? 
of  54,  63,  81  ?  of  21,  35,  84  ?      Ans.  252, 1134,  420. 

16.  "\^niat  is  the  least  common  multiple  of  Q6,  143, 
55  ?  of  144,  196, 128  ?  of  18,  45,  63  ? 

Ans.  4290,  56448,  630. 


102  PROPERTIES    OF   INTEGRAL    NUMBERS. 

17.  What  is  the  least  common  multiple  of  20,  35,  80  ? 
of  16,  24,  56  ?  of  26,  39,  65  ?       Ans.  560,  336,  390. 

18.  What  is  the  least  common  multiple  of  34,  51,  85  ? 
of  57,  95,  133?  of  69,115,161? 

Ans.  510,  1995,  2415. 

19.  What  is  the  least  common  multiple  of  8,  7,  10, 
14  ?  of  2,  6,  7,  29  ?  of  14,  21,  35,  49  ? 

Ans.  280,  1218,  1470. 

20.  What  is  the  least  common  multiple  of  272,  238, 
204,  170  ?  Ans.  28560. 

21.  What  is  the  least  common  multiple  of  11, 12,  13, 
14,  15,  16,  17,  18,  19,  20  ?  Aiis.  232792560. 

134.  To  find  the  least  common  multiple  of  two 
numbers, 

(1.)    Find  the  greatest  common  divisor  of  the  numhei^s. 

(2.)  Divide  one  of  the  members  by  this  divisor,  and 
multiply  the  quotient  by  the  other. 

The  product  will  be  the  least  common  multiple  of  the 
numbers. 

EXAMPLES. 

1.  What  is  the  least  common  multiple  of  1903  and 
4325  ? 

OPERATION. 


1903 

1557 

346 
346 

0 


4325 

3806 

519 
346 

173 


Then,  1903-v-173x4325-:47575  Ans, 
Or,      4325-^173Xl903--47575  Aris. 


PROPERTIES    OF   INTEGRAL    NUMBERS.  103 

2.  What  is  the  least  common  multiple  of  3471  and 
1869  ?— (Vide  122,  Ex.  9.)  Ans.  24297. 

3.  What  is  the  least  common  multiple  of  1137  and 
9475  ?— (Vide  122,  Ex.  10.)  Ans.  28425. 

4.  What  is  the  least  common  multiple  of  3447  and 
9575?  J.7i.§.  86175. 

5.  What  is  the  least  common  multiple  of  2359  and 
8425  ?  Ans.  58975. 

6.  What  is  the  least  common  multiple  of  117869  and 
137773?  *      ^ris.  52215967. 

125.   Practical  Applications. 

1.  The  diJBference  between  two  numbers  is  7,  and  the 
less  number  is  25.     What  is  the  greater?       Ans.  32. 

2.  The  difference  between  two  numbers  is  19,  and  the 
less  number  is  43.     What  is  the  greater?       Ans.  62. 

3.  The  difference  between  two  numbers  is  184,  and 
the  less  is  325.     What  is  the  greater?  Ans.  509. 

4.  The  difference  between  two  numbers  is  7,  and  the 
greater  is  32.     What  is  the  less?  Ans.  25. 

5.  The  difference  between  two  numbers  is  19,  and  the 
greater  is  62.     What  is  the  less?  Ans.  43. 

6.  The  difference  between  two  numbers  is  184,  and 
the  greater  is  509.     What  is  the  less?  Ans.  325. 

7.  The  difference  between  two  numbers  is  7,  and  the 
less  is  25.     What  is  their  sura?  Ans.  57. 

8.  The  difference  between  two  numbers  is  19,  and 
the  less  is  43.     What  is  their  sum  ?  Ans.  105. 

9.  The  difference  between  two  numbers  is  184,  and 
the  greater  is  509.     What  is  their  sum  ?       Aiis.  834. 


104  PROPERTIES    or    INTEGRAL    NUMBERS. 

10.  The  sum  of  two  numbers  is  105,  and  tlieir  differ- 
ence 19.     What  are  the  numbers  ?      Ans.  62  and  43. 

(1.)    Add  the  difference  to  the  sum,  and  divide  by  2. 

(2.)  Subtract  the  difference  from  the  sum,  and  divide 
by  2. 

The  result  will  be  the  numbers.     Thus, 

(1.)  (2.) 

105  105 

19  .  19 


2)124  2)  86 

62=Greater.  43:^Lcss. 

11.  The  sum  of  tAvo  numbers  is  5,  and  their  difference 
1.4.     What  are  the  numbers  ?  Ans.  3.2  and  1.8. 

12.  The  sum  of  two  numbers  is  783,  and  their  difference 
141.     What  are  the  numbers  ?  A71S.  462  and  321. 

13.  The  sum  of  two  numbers  is  46.4,  and  the  greater 
is  29.3.     What  is  the  less?  Ans.  17.1. 

14.  There  are  ^47.32  in  two  boxes.  One  of  them 
contains  §24.85.     How  much  money  in  the  other? 

Ans.  §22.47. 

15.  I  have  §323.67  in  two  purses.  One  of  them 
contains  §125.63.     How  much  in  the  other? 

Ans.  §198.04. 

16.  I  have  two  purses.  One  of  them  contains 
§198.04,  which  is  more  money  than  is  in  the  other  by 
§72.41.     How  much  does  it  contain  ?       Ans.  125.63. 

17.  My  money  is  in  two  purses,  both  of  which  con- 
tain §323.67,  and  there  are  §72.41  more  in  the  one  tlian 
in  the  other.     How  many  dollars  in  each  purse  ? 


PROPERTIES  •  OF   INTEGRAL   NUMBERS.  105 

18.  Two  men  together  own  3521.25  acres  of  land, 
but  one  of  them  owns  45.75  acres  more  than  the  other. 
How  many  acres  does  each  man  own  ? 

Ans.  1783.5  and  1737.75. 

19.  If  to  a  certain  number  I  add  45,  the  sum  will  be 
223.     What  is  the  number?  Ans.  178. 

20.  If  from  a  certain  number  I  take  34,  the  remain- 
der will  be  213.     What  is  the  number  ?        Ans.  247. 

21.  If  to  a  certain  number  I  add  14,  and  then  sub- 
tract 75  the  result  will  be  268.     What  is  the  number? 

Ans.  329. 

22.  If  from  a  certain  number  I  subtract  123,  and 
then  add  329,  the  result  will  be  930.  What  is  the  num- 
ber ?  Ans.  724. 

23.  The  divisor  of  a  number  is  45,  and  the  quotient 
is  73.     What  is  the  dividend?  Ans.  3285. 

24.  The  divisor  of  a  number  is  73,  and  the  quotient 
is  45.     What  is  the  dividend  ? 

25.  The  dividend  is  3285,  and  the  quotiei\t  is.  73. 
What  is  the  divisor  ? 

26.  The  dividend  is  3285,  and  the  quotient  is  45. 
What  is  the  divisor? 

27.  The  multiplier  is  45,  and  the  \product  3285. 
What  is  the  multiplicand? 

28.  The  multiplicand  is  45,  and  the  product  3285. 
What  is  the  multiplier  ?  •  .^ 

29.  If  a  certain  number  is  divided  by  321^  the  quo- 
tient will  be  23.     What  is  the  number  ?     ,  .  ^ 

30.  If  a  cej^tain  number  is  multiplied  by  321,  the 
product  will  be  7383.     What  is  the  number? 

31.  If  a  certain  number  is  multiplied  by  4,  and  the 


106  PllOPERTIES    OF   INTEGRAL   NUMBERS. 

product  is  then  divided  by  7,  the  quotient  will  be  16. 
What  is  the  number  ?  A^is.  28. 

32.  If  a  certain  number  is  divided  by  7,  and  the  quo- 
tient is  then  multiplied  by  4,  and  the  product  increased 
by  46,  the  sum  diminished  by  37,  the  result  will  be  25. 
What  is  the  number? 

33.  If  to  a  number  you  add  65,  and  from  the  sum 
subtract  38,  divide  the  diiference  by  2,  multiply  the 
quotient  by  3,  the  result  will  be  141.  What  is  the 
number?  Ans.  67. 

34.  The  dividend  is  251,  the  divisor  13.  What  is  the 
remainder  ? 

35.  The  dividend  is  251,  the  quotient  19,  and  the 
remainder  4.     What  is  the  divisor  ? 

36.  The  divisor  is  13,  the  quotient  19,  the  remainder 
4.     What  is  the  dividend  ? 

37.  ^he  divisor  is  13,  the  dividen4  251,  the  remain- 
der 4.     What  is  the  quotient  ? 

3^  Wliat  is  the  least  number  of  marbles  that  can  be 
divided  eqiially  among  2,  3,  4,  or  6  boys?         Ans.  12. 

39.  What  is  the  least  number  of  dollars  that  can  be 
divided  equally  among  8,  14,  or  21  men?      Ans.  168. 

40.  A  can  dig  7  rods  of  ditch  per  day;  B  caia  dig  13 
rods,  and  C  14  rods  in  the  same  time.  Wjbat  isAe  least 
number  of  rods  that  will  make  a  number^of  JBl  day's 

^ork  for  each  of  the  three  men?  '*     f-^^*  ■^^•2* 

41.  A/gentleman  has  145  gallons,  of  /ftatawbst',  203 
gallons  ^  Madeira,  and  319  gallons  of  Sci^ppjernong, 
and  he  desires,  to  fill  a  number  of  casks  of  equal  size 
without  mixing  or  wp^ting  the  wine.  How  many  gallons 
must  each  cask  hold  (—(Vide  121.)         Ans.  29  or  1. 


PROPERTIES    OE    INTEGRAL   NUMBERS.  107 

42.  A  farmer  has  482  bushels  of  corn,  622  bushels  of 
wheat,  and  758  bushels  of  barley.  He  wishes  to  fill  a 
number  of  sacks  of  equal  capacity,  not  mixing  the  grain, 
or  leaving  any  out.  IIo^y  many  bushels  must  each  sack 
hold?  Ans.  1  or  2  bushels. 

43.  A  merchant  has  64  silver  dollars,  72  half  dollars, 
and  144  quarters.  He  wishes  to  place  an  equal  number 
of  each  in  several  drawers,  not  mixing  them,  or  leaving 
any  out.  What  is  the  least  number  of  drawers  that  will 
answer  the  purpose  ?  Ans.  35. 

44.  A  certain  number,  on  being  divided  by  11,  12, 
13,  14,  15,  16,  17,  18,  19,  and  20,  respectively,  has  a 
remainder  on  each  division  just  one  less  than  the  divisor. 
What  is  the  number  ?— (Vide  123,  Ex.  21.) 

Ans.  232792559. 

45.  A,  B,  C,  and  D  start  together,  and  travel  the 
same  way,  round  an  island  500  miles  in  circumference. 
A  goes  8  miles  an  hour,  B  12,  C  16,  and  D  20.  What 
is  the  least  number  of  hours  that  will  bring  them  to- 
o;ether  aoiain  ?  Ans.  125. 

How  many  times  round  the  island  will  each  have 
traveled?  Ans.  A  2,  B  3,  C  4,  and  D  5  times. 

46.  Suppose  three  railroad  trains  to  start  at  the  same 
time  from  Quito,  and  to  run  round  the  earth  on  the 
equator  at  the  rate  of  3G^  60,  and  75  miles  an  hour  re- 
spectively. What  is  the  least  number  of  days  in  which 
all  will  arrive  at  Quito  at  the  same  time  ? — (Vide  96, 
Ex.  14.)  Ans.  69.1608  days. 

How  many  times  round  the  earth  will  each  train  have 
gone?        Ans.  First  2,  second  4,  and  third  5  times. 


108  FRACTIONS. 


FRACTIONS 


NATURE   OF   FRACTIONS. 

126.  If  an  apple  is  divided  into  two  equal  parts,  each 
part  is  said  to  be  a  half  of  the  whole  apple. 

If  an  orange  is  divided  into  tJwee  equal  parts,  each 
part  is  said  to  be  a  third  of  the  whole  orange. 

If  a  line  is  divided  into  four  equal  iJarts,  each  part  is 
said  to  be  di,  fourth  of  the  whole  line. 

127.  If  any  quantity  whatever  is  divided  into  a  given 
nuraher  of  equal  parts,  each  pai^t  takes  a  name  lohich 
indicates  the  nurdher  of  parts  into  tuhich  the  quantity  is 
divided.     Thus, 

A  half  indicates  a  division  into  two  equal  parts. 

A  third  indicates  a  division  into  three  equal  p>arts. 

A  fourth  indicates  a  division  into  four  equal  p)arts. 

A  fifth,  sixth,  seventh,  eighth,  ninth,  etc.,  indicate, 
when  applied  to  any  quantity,  that  it  has  been  divided 
into  five,  six,  seven,  eight,  nine,  etc.,  equal  parts. 

12S.  If  an  apple  is  divided  into  any  number  of  equal 
parts,  each  part  is  a  whole  part  or  unit,  and  the  word 
one  may  therefore  be  applied  to  it.  (Vide  4.)  Thus, 
one  half,  one  third,  one  fourth,  one  fifth,  one  tenth,  one 
twentieth,  etc. 

More  than  one  part  may  be  indicated,  just  as  more 
than  one  of  any  other  quantity  is  indicated.  Thus,  two 
thirds,  two  fourths,  three  fourths,  two  halves,  four 
fourths,  seven  tenths,  nineteeii  twentieths,  etc.,  any  of 
which  expressions  is  called  a  fraction.     Hence, 


FRACTIONS.  109 

(1.)  A  FRACTION  represents  one  or  more  than  one  of 
the  equal  parts  of  a  unit. — (Vide  8.) 

(2.)  The  UNIT  OF  A  FRACTION  is  the  whole  quantity 
from  which  the  fraction  is  derived. 

(3.)  A  FRACTIONAL  UNIT  is  ONE  of  the  equal  parts  of 
the  whole  quantity  that  is  divided. — (Yide  38,  I.) 

NOTATION    OF    FRACTIONS. 

120.  The  unit  of  a  fraction  may  always  be  repre- 
sented by  the  figure  1. 

130.  A  fractional  unit  is  represented  by  drawing  a 
horizontal  line  under  the  figure  1,  and  placing  under- 
neath it  the  figure  denoting  the  number  of  parts  into 
which  the  unit  of  the  fraction  is  divided.     Thus, 

One  half  is  represented  by J. 

One  third  is         "  "        -J. 

One  fourth  is       "  "        J. 

One  fifth  is  "  "        i- 

One  sixth  is         "  "        %. 

One  seventh  is     "  "        4- 

One  twentieth  is "  " .  j*^. 

One  eighty-fifth  is  represented  by -i^. 

Remark. — The  reciprocal  of  a  number  is  represented  by  placing 
the  number  under  the  figure  1  in  the  manner  of  a  fractional  unit. 
Thus,  the  reciprocal  of  2  is  ^;  of  3  is  \]  of  100  is  j-^^,  etc. 

131.  Any  given  number  of  fractional  units  is  repre- 
sented by  writing  the  given  number  above  the  line  in 
place  of  the  figure  1.     Thus, 

Two  thirds  is  represented  by |. 

Two  fourths  is         "  " f . 

Three  fo\irths  is       "  " f . 


110 


FRACTIONS 


23 

S5' 


Two  fifths  is  represented  by 

Three  fifths  is  "         " 

Four  fifths  is  "         " 

Seven  twentieths  is  "         " 

Twenty-three  eighty-fifths  is  represented  by 

XS2.  Properly  a  fraction  represents  a  less  number  of 
fractional  units  than  is  contained  in  the  unit  of  the 
fraction.  The  unit  of  the  fraction  may,  however,  be 
represented  in  the  form  of  a  fraction,  by  writing  that 
figure  above  the  line  which  denotes  the  entire  number 
of  fractional  units  contained  in  it.  Thus,  two  halves 
is  represented  by  §,  three  thirds  by  |,  etc. 

133.  A  greater  number  of  fractional  units  than  is 
contained  in  the  unit  of  the  fraction,  may  also  be  repre- 
sented in  the  form  of  a  fraction.     Thus, 

Three  halves  is  represented  by §. 

Four  halves  is  "  "......     4 

Five  halves  is 
Four  thirds  is 
Five  thirds  is 
Six  thirds  is 
Seven  thirds  is 
Seventeen  sixths  is 

134.  An  integral  number  is   jo 
thus,  (vide  92,  Rem.  2,) 
One  and  one  half  is  represented  by 
One  and  one  third  is  represented  by 
Two  and  one  half  is  "  " 
Three  and  one  fifth  is        "           " 
Five  and  three  fifths  is      "           " 
Eight  and  six  thirteenths  is  represented  by 


ned   to  a  fraction 


21. 

5|. 
8A- 


FRACTIONS.  Ill 

135.  When  one  quantity  is  to  be  divided  by  another, 
the  dividend  and  divisor  may  be  written  in  the  form  of 
a  fraction.     Thus, 

Seven  divided  by  five  is  represented  by      ....     J 

One  half  divided  by  three  is  represented  by    ...     | 


One  divided  by  one  half  is  represented  by      •     •     •     i 
One  third  divided  by  one  fifth  is  represented  by  .     .     f 

3" 

2 

Two  divided  by  five  and  one  third  is  represented  by     gy 
Two  and  one  half  divided  by  three  and  one  third  is 

represented  by rf 

136.  Since  any  fractional  unit  is  a  whole  part  of  the 
unit  of  the  fraction,  it  may  itself  he  divided  into  any 
number  of  equal  parts,  and  each  part  into  any  number  of 
other  equal  parts,  and  so  on  to  any  extent  whatever. 
Thus,  a  half  dollar  may  be  divided  into  two  equal  parts, 
and  each  of  these  into  five  other  equal  parts,  etc. 

This  division  is  indicated  by  figures,  thus : 
One  half  of  one  half  is  indicated  by      ...     J  of  J. 
One  fifth  of  one  half  is  "         "...     i  of  J. 

One  fifth  of  one  seventh  is    "         "        ...     i  of  4. 
One  fifth  of  one  third  of  one  fourth  is  indicated 

i>y J  of  J-  of  1. 

NUMERATOR  AND  DENOMINATOR. 

137.  In  every  fraction,  the  figure  above  the  line 
indicates  the  number  of  fractional  units  taken.  It  is 
thence  called  the  numerator  of  the  fraction. 


112  FK  ACTIONS. 

138.  In  every  fraction  the  figure  below  tlie  line  in- 
dicates the  number  of  equal  parts  into  which  the  unit  of 
the  fraction  is  divided.  It  therefore  determines  the 
name  to  be  applied  to  the  fractional  unit,  and  is  thence 
called  the  denominator  of  the  fraction. 

Remark  1. — The  numerator  and  denominator  taken  together  are 
called  the  terms  of  a  fraction. 

Remark  2. — The  terms  of  a  fraction  are  said  to  be  inverted  when 
the  numerator  takes  the  place  of  the  denominator,  and  the  reverse. 
Thus,  the  fraction  f  inverted  becomes  |. 

Remark  3. — Every  fraction  is  said  to  be  in  its  lowest  terms  when 
they  are  prime  with  respect  to  each  other. — (Vide  102.) 

Remark  4. — When  a  quantity  is  written  in  the  form  of  a  fraction^ 

the  part  above  the  principal  line  is  still  called  the  numerator^  and 

1 
that  below,  the  denominator.     Thus,  in  the  expression  | ,  one  half  is 

the  numerator,  and  one  third  the  denominator. 

CLASSIFICATION  OF  FRACTIONS. 

139.  A  Simple  Fraction  is  one  whose  terms  are  each 
single  integral  numbers.  Thus,  J,  |,  f,  f,  y,  |J,  etc.,. 
are  simple  fractions. 

140.  A  Proper  Fraction  is  a  simple  fraction  whose 
numerator  is  less  than  the  denominator.  Thus,  J,  |,  f , 
?>  A?  IJj  ^*^-?  ^^®  proper  fractions. — (Vide  132.) 

141.  An  Improper  Fraction  is  a  simple  fraction  whose 
numerator  is  equal  to,  or  greater  than,  the  denominator. 
Thus,  I,  I,  I,  f,  f,  V?  if  J  ^tc.,  are  improper  fractions. — 
(Vide  133.) 

142.  A  Compound  Fraction  is  an  expression  consist- 
ing of  two  or  more  simple  fractions,  united  by  the  word 
OF.  thus,  I  of  i,  I  of  J,  \  of  I  of  i  of  I  of  f,  are 
compound  fractions. — (Vide  136.) 


FRACTIONS.  '  113 


143.  A  Mixed  Fraction  is  an  integral  number  united 
to  a  proper  fraction.  Thus,  1^,  Ij,  2^,  31,  8^^,  9^1, 
etc.,  arc  mixed  fractions. — (Vide  134.) 

144.  A  Complex  Fraction  is  a  quantity  written  in 
the  form  of  a  fraction,  in  which  one  of  its  terms  is  a 
fraction,  or  both.  Thus,  tt^  v?  vjvt,^,^^  are  complex 
fractions. — (Vide  135.) 

VALUE   OF   A   FRACTION. 

145.  The  value  of  a  fraction  is  the  quotient  of  the 
numerator  divided  by  the  denominator. 

Remark  1. — The  value  of  a. proper  fraction  is  less  than  1. 

Remark  2. — The  value  of  an  improper  fraction  is  equal  to  or 
greater  than  1.     Thus,  f=l;  |=1 ;  f=l^;   V=2j  etc. 

Remark  3. — The  value  of  a  fraction  is  such  a  part  of  the  nu- 
merator as  is  denoted  by  the  reciprocal  of  the  denominator.  Thus, 
1=1  of  5;  -|=i  of  4;  -|=i-  of  4;  |=i  of  5,  etc.— (Vide  127  and 
130,  Rem.)  " 

PROPOSITIONS. 

I.  The  value  of  a  fraction  is  not  changed  by  multi- 
plying or  dividing  both  terms  by  the  same  number. 

II.  The  value  of  a  fraction  is  multiplied  when  the 
numerator  is  multiplied  or  denominator  divided. 

III.  The  value  of  a  fraction  is  divided  when  the  nu- 
merator is  divided  or  denominator  multiplied. 

REDUCTION  OF  FRACTIONS. 

140.    The  reduction  of  a  fraction  consists  in  changing 
its  form,  or  the  value  of  its  terms,  without  altering  the 
value  of  the  fraction. 
10 


114 


FRACTIONS 


147.    To  reduce  a  simple  fraction  to  its  lowest  terms, 
Divide  the  numerator  and  denominator  hy  their  greatest 
common  divisor ,  or  cancel  the  factors  common  to  the  nu- 
merator and  denominator. 

EXAMPLES. 

1.  Reduce  f  to  its  lowest  terms.  Ans.  J. 

2.  Reduce  §,  f ,  ^%,  |,  /^-,  Jg,  /^,  and  >f ,  each  to  its 
lowest  terms.  Ans  -f ,  f,  |,  |,  |,  f,  J,  and  f . 

3.  Reduce  f  J,  f§,  if  J,  |||,  if^  ||,  and  |f,  each  to 
its  lowest  terms.  Arts.  J,  f ,  |,  g,  ^,  f ,  and  y%. 


OPERATION. 

(Vide  115.)     9|i-||=ig==-Mn5. 

4.   Reduce  H,  -j3^<V,  ^^(j.  JS.  IS.  ??,  and  ^%%,  each  to 
its  lowest  terms.  Ans.  |,  f,  |,  |,  f,  |,  and  f. 

^       "Rprlnpp    18     105       84         9_9        78      117    o^lfl   19^    paph 

o.    xveuute  -3  q,  175?  T40'  les?  Tsn?  Tirs?  '^^^^  325?  ^^^^ 
to  its  lowest  terms.  Ans.  f. 

6"RprInf»P    IT      3  8      6  9      116      155      J2  5  9      104     nQoli+nif^ 
.    xieauce  -34,  ^t^,  ^^j  T45j  ths?  296?  1  it?  ^^^^^  ^^  ^^^ 

lowest  terms.  Ans.  J,  |,  |,  etc. 

7       PprlnPA     39      51        69         87        169      183       213       popVl     tn 

*•    xieauce  g^,  g-g,  yyg,  y45,  25^?  sTJSJ  355>  ^^^^i  ^^ 
its  lowest  terms.  Ans.  |. 

8.  Reduce  f  ||g  to  its  lowest  terms. 

OPERATION. 

(Viae  iiy,  j^x.  iv.;     4^2  0—2x2x3X5X7X11     ' ' 

Remark. — In  practice  a  line  may  be  drawn  across  the  common 
factors,  and  the  factors  in  each  term  not  crossed  must  be  multi- 
plied together.     Thus, 

9.  Reduce  -f  f  g  to  its  lowest  terms. 


FRACTIONS.  115 

OPERATION. 
,4  6_^X3>aXl3_3  9         A^ 

10.  Reduce  J-JiJ  to  its  lowest  terms.  Ans.  {J. 

11.  Reduce  ?J|J  to  its  lowest  terms.  Aiis  ^|f. 

12.  Reduce  jii  J  to  its  lowest  terms.  A')is,  -J|. 

13.  Reduce  Iff  §  to  its  lowest  terms.  Ans.  [f. 

14.  Reduce  Jitlfl  to  its  lowest  terms. 

OPERATION. 

(Vide  122,  Ex.  11.)     3ii|J.J.|B..|^ 

15.  Reduce  Iff  ?  to  its  lowest  terms. 

16.  Reduce  -JJ||  to  its  lowest  terms. 

17.  Reduce  J|§f  to  its  lowest  terms. 

18.  Reduce  |f-|J  to  its  lowest  terms. 

19.  Reduce  ||||  to  its  lowest  terms. 

20.  Reduce  }  J||  and  j%\%  to  their  lowest  terms. 

Ans.  hi  and  ||. 

148.  To  reduce  a  simple  fraction  to  another  fraction 
having  a  given  denominator, 

(1.)    If  necessary,  reduce  the  fraction  to  its  lowest  terms. 

(2.)  Divide  the  proposed  denominator  by  the  denomi- 
nator of  the  reduced  fraction. 

(3.)  Midtiply  both  terms  of  the  reduced  fraction  by  the 
quotient. 

EXAMPLES. 

1.  Reduce  \l\%  to  a  fraction  whose  denominator 
shall  be  69. 

OPERATIONS. 
(1.)  (2.)  (3.) 

—  23X3       ^^  ^* 


379        J^n 

44  3    -^^ 

*  ■ 

Ans. 

t\- 

Ans. 

s^- 

Ans. 

M- 

Ans. 

A- 

Ans. 

2\- 

116 


FRACTIONS. 


2.  Reduce  y°o   to  fractions  whose  denominators  shall 
be  9, 15, 18,  21,  and  27.        Ans.  f ,  J  g,  jf,  if,  and  Jf. 

3.  Reduce  !§  to  fractions  whose  denominators  shall 
be  15,  25,  30,  35,  40,  45,  etc. 

477.9      -9-      J  S       18      2  1      pf« 

4.  Reduce  f,  ;|,  g,  and  /q,  to  fractions  whose  denomi- 
nators shall  each  be  60.  Ans.  |J,  |g,  |g,  and  |f. 

5.  Reduce  |,  Z^,  2§5  ^'^i^d  i\?  t<^  fractions  whose  de- 
nominators shall  each  be  30.   A71S.  fg,  §J,  -Jf,  and  -Ig. 

6.  Reduce  |J,  J|,  y'/g,  and  J|,  to  fractions  whose 
denominators  shall  all  be  105.        Ans.  /q\,  -^^-g,  etc. 

7.  Reduce  f  i,  4§,  Jf  I'  ^^^^  iii'  *^  fractions  whose 
denominators  shall  be  280.  Ans.  Jl^,  etc. 

8.  Reduce  -J|,  ||,  ||,  and   [if,  to  fractions  whose 
denominators  shall  be  60.  Ans.  |J,  etc. 

9.  Reduce  f |,  iff,  -|f },  and  ||f,  to  fractions  whose 
denominators  shall  be  504.  Ans.  |§J,  etc. 

10.  Reduce  Jf  |,  o^,  f  |,  and  /g^,  to  fractions  whose 
denominators  shall  be  126.  Ans.  j^^^,  etc. 

a  fraction  whose  denominator 

Ans.  If. 
a  fraction  whose  denominator 
Ans.  \2. 
to  fractions  whose  denominators  shall 
Ans.  f,  y,  y,  etc. 

14.  Reduce  13, 14,  15,  16,  17,  18,  19,^  and   20   to 
fractions  whose  denominators  shall  all  be  17. 

Ans.  ^iV>  ¥-7%  etc. 

15.  Reduce  11, 12,  21,  22,  23,  24,  and  25  to  fr^- 
tions  whose  denominators  shall  be  19. 

Ans.  Yg%  t/>  etc. 


11.   Reduce  Jff J 

to 

shall  be  52. 

12.   Reduce  3=f 

to   ; 

shall  be  4. 

13.   Reduce  5  to 

fraci 

be  1,  2,  3,  4,  5,  6,  7, 

etc. 

FRACTIONS.  117 

16.  Reduce  19  to  fractions.Tvhose  denominators  shall 
be  11,  12, 13,  14,  etc.,  to  20.  Ans.  \\9,  ^^^^  etc. 

17.  How  many  half  dollars  in  23  dollars  ? 

Ans.  46  half  dollars. 

18.  How  many  quarters  in  23  dollars  ? 

Ans.  92  quarters. 

149.  To  reduce  a  mixed  number  to  an  improper 
fraction  which  shall  be  in  its  lowest  terms, 

(1.)  Reduce  the  fractional  part  to  its  lowest  terms,  if  it 
is  not  so  given. 

(2.)  Multiply  the  whole  number  by  the  denominator  of 
the  reduced  fraction,  and  add  to  the  product  the  nu- 
merator. 

(3.)    Write  the  sum  over  the  reduced  denomiiiator. 

EXAMPLES. 

1.  Reduce  23j|fg,  15ff§g,  and  7|4§,  to  improper 
fractions  in  their  lowest  terms. — (Vide  147,  Ex.  9  and 
10;  also,  148,  Ex.  1.) 

OPERATIONS. 

(1.)  (2.)  (3.) 

23i§!S-23H  154fiS-15H  7f4g=r|f 

SgV    Ans.  Y_6     J^^jg.  4_2_4    ^^^s. 

2.  Reduce  13f,  15|,  17|,  and  I^^q,  to  improper 
fractions  in  their  lowest  terms. 

Ans.  y,  V?  V^  and  V- 
S.   Reduce  21f ,  23/o,  27i§,  and  31|f,  to  improper 
fractions  in  their  lowest  terms. 

J/*,<?     6  5      7  1       13  8      ori,l    1  27 

JLns.    3,3,     5    ,  aiiu     ^   . 


118  FRACTIONS. 

4.  Reduce  17j  J,  16||,  12f  |,  and  19;  J  |,  to  improper 
fractions  in  their  lowest  terms. 

Ans.  %^,  y^,  V,  and  %K 

5.  Reduce  14||,  15|i,  IGyVs?  and  ITyVs,  to  improper 
fractions  in  their  lowest  terms.  Aiis.  \^,  ''/,  etc. 

6.  Reduce  lSj%,  ITy^,  19tV,  and  2d.^\,  to  improper 
fractions  in  their  lowest  terms.  A7is.  \y,  etc. 

7.  Reduce  124jf|,  256/r\^  211|ff-g,  and  112j-||i, 
to  improper  fractions  in  their  lowest  terms. 

J^a     1120     1795     3600     and^^^^ 

150.  To  reduce  an  improper  fraction  to  an  integral 
or  mixed  number, 

Divide  the  numerator  hy  the  denominator,  and  place 
the  excess  of  fractional  units  to  the  right  of  the  quotient. 

Remark. — Let  it  be  understood  that,  unless  special  direction  is 
given  to  the  contrary,  all  answers  are  to  be  given  in  their  lowest 
terms. 

^-dB-l^A^FL  E  S  . 

1.   Reduce  '4'*  to  an  improper  fraction.        A71S.  3^. 

2.  Reduce  |,  f,  |,  y ,  y,  y,  ^/,  to  integral  or  mixed 
numbers.  Ans.  4,  4,  4.^,  2^,  2i,  3,  3]. 

3.  Reduce  \4,  V?  'P?  'i¥'  ^V^  ^^^d  \8^,  to  mixed 
numbers.  Ans.  13i,  15|,  17|,  18|,  21|,  23f . 

4.  Reduce  ^^f,  YsS  W?  ¥?  Wj  to  mixed  numbers. 

J.WS.  23^1,  etc. 

5.  Reduce  1//,  ^^{,  4_y,  \¥?  and  1/^2^  to  integral 
numbers.  Ans.  13,  14,  etc. 

6.  Reduce  W>  W?  ¥eS  \V>  ¥2S  to  mixed  num- 
bers. Ans.  ISJ^,  I7J5,  20/g,  20  Jj,  20i. 

7.  Reduce  W?  \¥j  W?  \V?  ^^^  W?  to  mixed 
numbers.  j-ws.  19, \,  etc. 


FRACTIONS.  119 

8.  Reduce  -%^^,  '"'^S^^,  ^V/^  and  5f|f^  to  mixed 
numbers.  Ans.  3574,  1228^^,  etc. 

ADDITION   OF   FKACTIONS. 

151.    To  add  two  or  more  proper  fractions  together, 

(1.)  Reduce  the  fractions  to  their  lowest  terms ^  if  they 
are  not  so  given. — (Vide  147.) 

(2.)  Find  the  least  common  multiple  of  the  reduced 
denominators, — (Vide  123.) 

(3.)  Reduce  each  fraction  to  one  which  shall  have  a 
denominator  denoted  hy  the  least  common  multiple. — (Vide 
148.) 

(4.)  Add  the  numerators  of  the  resulting  fractions,  and 
if  the  sum  placed  over  the  denominator  is  an  improper 
fraction,  reduce  it  to  an  integral  or  mixed  number. — 
(Vide  150.) 

Remark. — If  the  given  fractions  all  have  the  same  denominator, 
their  numerators  should,  of  course,  be  added  at  once  by  (4.) 

EXAMPLES. 

1.  Add  together  f ,  |,  |,  and  f^. 

OPERxVTION. 

f+i+i+A  =Given  fractions. 

i+l+f+l  (Vide  147,  Ex.  1  and  2.) 
(Vide  148,  Ex.  4.)  iHiHiS+ig=  W^Sfi  Ans, 

2.  Add  together  f ,  f^,  J  §,  and  ^-^.—(yide  148,  Ex. 
5.)  ^  Ans.  2t3. 

3.  Add  together  | J,  Jf .  iVa.  and  J]-.     Ans.  2-f^%. 

4.  Add  together  if,  fj-,  -,\%,  and  ^..       Ans.  2f. 

5.  Add  together  §,  f  i,  |-f,  and  |i.  Ans.  IJ. 


120  FRACTIONS. 

6.  Add  together  §,  y\,  -^^^  and  ^V-       ^'^s.  Ijoo- 

7.  Add  together  A  8,  1.0  5;_8_4_^  and /gV     -^^s.  2|. 

8.  Add  together  4,  f ,  f ,  and  4-  -^^s.  If. 

9.  Add  together  -J,  f,  -|,  and  |.  ^ws.  1-J. 

10.  Add  together  I,  J,  i,  and  J.  Ans.  l^J. 

11.  Add  together  J,  |,  f ,  and  |.  ^ns.  2jf . 

12.  Add  together  |,  -/g,  |,  and  j4_^.  JlWS.  2/^. 

13.  Add  together  4,  Z^,  ^\,  and  3%-.         ^tis.  Jf  g. 

14.  Add  together  f |,  i?-f ,  ifi,  and  -|if. 

^/^§^  3229^ 

15.  Add  together  h^,  /,%,  ff,  and  ^%\. 

Ans.  1{^. 

16.  Add  together  |,  -j^,  4,  f ,  and  ^^g.       J.?zs.  2/^. 

17.  Add  together  y/^g  and  ^Z^^.— (Vide  124,  Ex. 
1.)  Ans.  44J45. 

18.  Add  together  -^^^j  and  yj§^.         ^tis.  of  oIt-. 
19   Add  together  y  1^377  and  9  yV?-  ^ns.  5/4^0  5. 

Kemark. — When  mixed  numbers  are  required  to  be  added,  add 
the  sum  of  the  fractional  parts  to  the  sum  of  the  integral  numbers. 

20.   Add  together  3j  and  4j;  also,  7j  and  6| ;  also, 
14|  atid  135. 

OPERATIONS. 

(1.)  (2.)  (3.) 


3i= 

=3^ 

n=n 

14i=14|f 

H= 

=4A 

^=^ 

13f=13|§ 

7A 

Ans. 

14i  Ans. 

28|f  Ans. 

21. 

Add  together  13f  and  14|. 

Ans.  28J. 

22. 

Add  together 

171  and  18  fV 

Ans.  36iJ. 

23. 

Add  together 

21f  and  23,8^-. 

Ans.  45 J. 

24. 

Add  together 

17||andl6|?. 

Aws.  34^. 

FRACTIONS.  121 

25.  Add  together  4j,  3j,  4i,  and  6|.     Ans.  18io  §. 

26.  Add  together  7^,  ISj^^-,  16|,  and  5j. 

Ans.  42f  3.. 

27.  Add  together  4i,  2i,  28|f,  and  29j-J. 


28.   Add  together  9i,  8^^,  7o^5,  and  65. 


^7^s.  64|. 


Ans.  89f f. 

29.  Add  together  1 J  and  2i ;  also,  Sj  and  4J ;  also, 
2|  and  Ij;  also,  Ij  and  2j. 

^"*'   ^15?     '"30?     "^Bf    ^40* 

30.  Add  together  4Jq  and  3^^;  also,  2  J^  and  5-J; 
also,  5J3  and  7^^.  Ans.  7/-  7/^;  12/^. 

31.  Add  together  2^  and  1 J^  5  ^^so,  3Jj  and  4J- ; 
also,  51  and  3J.  ^ns.  3|f ;  7if ;  8/0- 

32.  Add  together  3f  and  4f  ;  also.  If  and  5| ;  also, 
4|and6|.  Am.7U;  6fi;  llji. 

SUBTRACTION   OF   FRACTIONS. 

XS2.   To  subtract  one  proper  fraction  from  another, 

(1.)  Reduce  the  fractions  to  their  lowest  terms,  if  they 
are  not  so  given. 

(2.)  Find  the  least  common  multiple  of  the  reduced  de- 
nominators, and  reduce  each  fraction  to  the  denominator 
denoted  by  it. 

(3.)  Take  the  numerator  of  the  subtrahend  from  that 
of  the  minuend,  and  place  the  difference  over  the  denomi- 
nator.— (Vide  150,  Rem.) 


EXAMPLES. 

.   From  1 

take 

2\- 

2. 

From 

1% 

take 

1% 

11 

3. 

From 

Jl take 

hh 

^ 

>•"' 

- 

12-2  FRACTIONS. 


OPERATIONS. 

(1.)  (2.)  (8.) 

8  —  /f  1% — 1\  if — ii         Given  fractions. 


1.  .3-_i  13. 

3  8         4  1^ 


(Vide  147.) 
i-f-^  A718.  l-i=^iAns.  ||-i|-3^^^s.  (Vide  148.1 

4.  What  is  the  value  of  |-A?  of  |-|?  of  jl.—^^'i 

of  A-^r'  ^^^^-  8%;  T6;  §§;  f- 

5.  What  is  the  value  of  i— A?  of  J— J^V  of  ^—^J 

6.  What  is  the  value  of  y%— §'?  of  Ji  — J?  of  Ji— |? 

7.  What  is  the  value  of  /f— 1\?  of  |— ^?  of  4— _i-? 

of      6 189  J  ^,9      1.       7.     37.     1 

8.  From  {J.j  take  «|.  ^?«8.  J,. 

9.  From  f ||  take  f|9,  j^^s^  _i^^ 

10.  From  Hf  take  f|J.  ^^s.  1. 

11.  From  ^^%%  take  5%.  J.^s.  ^. 

12.  From  34^^  take  y^V?-  ^^s-  ^eVf 5- 

13.  From  ^g^^^  take  ^f^^.  A7is.  ^gV^^. 
14.'  From  j^%^  take  34^^,-.  J.ws.  ^ J/g^. 

15.  From  ^/g^  take  ^4^-5.  ^ws.  olfl^. 

Remark  1. — A  proper  fraction  is  taken  from  1  by  writing  the  dif- 
ference between  the  numerator  and  denominator  over  the  denominator. 
Thus,  1—3^1   since  f  —  |  =  1      (Vide  132.) 

And  1—^^=3^  since  \l—^^=3^ 

16.  From  1  take  §,  |,  J,  J,  a,  |,  f ,  |,  |,  and  f 

yd  790  1  1  4  1  2         p  +  p 

yiAts.  25  -3,  5,  2,  3,  eio. 

17.  From  1  take  J§,  /„  J|,  ,*t,  /„  Jg,  and  |J. 


^- 


18.   From  3  take  §,  |, 


Ans 

•5^0 

1  5 
?    2  2"? 

iJ, 

etc, 

1, 

4,i, 

5 

A, 

and 

sV 

4??s. 

Ol, 

01 

04 

25, 

etc, 

FRACTIONS. 


19.  Findthe  value  of  9— J§;  of  24— ^^.  ;  of37— '«; 
of  86-/^.  Ans.  8/^;  23||;  36 Ji;  85/^. 

20.  Find  the  value  of  4— f ;  of  121— J§ ;  of  187—^^  ; 
of  2145-IJ-.  Ans.  3|;  120,^^;  186,3^;  2144f. 

21.  From  8  take  24..  22.   From  4j  take  3J-. 


23.   From  5i  take  3 


OPERATIONS. 

(21.)  (22.)  (23.) 

8  4j-4i  5i=5§ 

51  Ans.  1|  Ans.  1|  Ans. 

The  first  two  examples  need  no  explanation.  In  the 
last,  having  reduced  the  fractional  parts  to  the  same 
denominator,  the  numerator  of  the  upper  fraction  is 
added  to  the  difference  hetiveen  the  terms  of  the  lower 
fraction  for  the  numerator  of  the  ansiver. 

This  is  the  same  as  adding  a  fractional  unit,  in  terms 
of  the  reduced  fraction,  to  the  upper  fraction,  and  then 
subtracting  the  lower  fraction.     Thus,  |+§ — i=i. 

The  true  difference  is  preserved  by  adding  1  to  the 
lower  integral  number,  before  taking  i.t  from  that  above 
it.     Hence, 

Remark  2. — Always  proceed  in  this  way  when  the  fractional 
part  of  a  mixed  subtrahend  is  greater  than  that  of  the  minuend. 

24.  From  28i  take  l^.   ■  Ayis.  14j 

25.  From  36ji  take  17f .  Ans.  18f 

26.  From  45|  take  23/^.  Ans.  21|. 

27.  From  34i  take  17J  J.  Ans.  16f . 

28.  From  123|  take  45/^.  Ans.  1^^^, 


4 

FRACTIONS. 

29. 

From  106|  take  103g. 

Ans.  3 J. 

30. 

From  165/j-  take  63^^. 

Ans.  102//^-. 

31. 

From  71Ji  take  3f  J. 

Ans.QlUi. 

32. 

From  19^-  take  lOff. 

Alls.  8|||. 

33. 

From  55j%  take  37i-gi. 

Ans.  17lil. 

34. 

From  25jV5  take  18|4|. 

Ans.  6i|. 

35. 

From  122 J  take  16 j|. 

Ans.  105f  f . 

36. 

From  1345f  take  237|. 

Ans.  1107f  i. 

37. 

From  1000  take  555  j\. 

A71S.   4:4:4:  ^^j. 

Remark  3. — Improper  fractions  may  be  subtracted  precisely  like 
proper  fractions,  but  it  is  generally  much  better  to  reduce  them  to 
mixed  numbers  before  subtracting.     Thus, 

.8 1  —  8 8—1 44—1 42      (Vide  Ex.  24.) 


MULTIPLICATION   OF   FRACTIONS. 

153.  To  multiply  a  simple  fraction  by  an  integral 
number, 

(1.)  Multiply  the  numerator  by  the  integral  number, 
and  plaee  the  product  over  the  denominator;  or,  (vide 
145,  II,)  _  .        . 

(2.)  Divide  the  denominator  by  the  integral  number,  if 
it  is  exactly  divisible,  and  place  the  quotient  under  the 
numerator. 

Remark  1; — All  answers  should  be  integral  numbers,  mixed 
numbers,  or  proper  fractions. — (Vide  150,  Rem.) 


EXAMPLES 


1.  Multiply  I  by  10.     Ans.  6. 


3.  Multiply  f  by  6. 

4.  Multiply  T^y  by  2. 


Ans.  1\. 
Ans.  fo-. 


6.  Multiply  Jy  by  5.  Ans.  2\. 

6.  Multiply  ^jy  by  4.  Ans.  1^. 

7.  Multiply  \l  by  15.  Ans.  ^. 

8.  Multiply  V  by  14.  Ans.  24. 


FRACTIONS.  125 

9.   Multiply  I  by  2,  3,  4,  5,  6,  7,  8,  9,  and  10. 

Ans.  If,  2|,  3i,  etc. 

10.  Multiply  1^  by  11,  12,  13,  14,  15,  16,  17,  18, 
and  19.  Ans.  7f^g,  8i,  etc. 

11.  Multiply  JJ  by  20,  25,  30,  35,  40,  45,  50,  and 
100.  Ans.  13f ,  17,  etc. 

Remark  2. — Any  simple  fraction  multiplied  by  its  denominator 
produces  the  numerator  for  a  product.     Thus, 

VX5-17;  f|fX163=2o9. 
154.    To  multiply  a  mixed  number  by  an   integral 
number, 

(1.)  Multiply  the  fractional  part  as  in  153. 
(2.)  Multiply  the  integral  part  as  in  80. 
(3.)  Add  the  products  together. 


EXAMPLES. 

1.   Multiply  4|  by  10.              2.    Multiply  7|  by  6. 

3.   Multiply  9/5  by  5 

• 

OPERATIONS. 

(1.)                         (2.) 

(3.) 

4f  (vide  Ex.  1, 153.)  7i  (vide  Ex.  3.) 
10                                6 

9/g  (vide  Ex.  5.) 
5 

46  Ans.                        44  i  Ans. 

47|  A71S. 

4.   Multiply  27/3  by  2. 

Am.  54}  J. 

i     5.    Multiply  29/0  by  4. 

Ans.  117|. 

6.    Multiply  121  §  by  15. 

Ans.  1831. 

7.   Multiply  15f  by  5. 

Ans.  78. 

8.   Multiply  13|  by  18. 

Ans.  250. 

9.   Multiply  14/^  by  34. 

Ans.  486. 

126  FRACTIONS. 

10.  Multiply  1311  by  11,  12,  13,  14,  15,  16,  17,  18, 
and  19.  Last  Ans.  260^^^. 

Remark, — A  complex  fraction  may  be  reduced  to  a  simple  frac- 
tion bj/  multiplying  both  terms  of  the  fraction  by  the  least  common  mul- 
tiple of  the  denominators  of  the  fractional  part s^. 

11.  Reduce  |,  -f,  and  ~^  to  simple  fractions. 

OPERATIONS. 

(1.)  (2.)  (3.) 

3X4     T^^^^-    6ix6-3^^^'^*    3^-3^X10-31-11  ^''^- 

•       1  1        1 1       1  -2_  33  '  < 

12.  Reduce  rf,  rf,  r|,  and  rf,  to  simple  fractions. 

^3       ^S      ^1  ^¥ 

^^s-  hh  A.  h  and  |. 

13.  Reduce  J,  r?  "?5  and  -_%  to  simple  fractions. 

J_?2S.  -/o5  14,  I,  and  ^%. 

14.  Reduce  |,  ?^,   |  |,  ^,  and  ^^,   to  simple 

f^^^ti^^l-  ^  ^  ^^..  4?,  il,  Y,  etc. 

155.  To  multiply  an  integral  number  by  a  simple 
fraction,  ' 

(1.)  Divide  the  integral  number  hy  the  denominator 
of  the  fraction,  and  multiply  the  quotient  hy  the  nu- 
merator ;   or, 

(2.)  Multiply  the  integral  number  by  the  numerator  of 
the  fraction,  and  divide  the  p)roduct  by  the  denominator. 


EXAMPLES. 

1.  Multiply  20  by  f . 

2.  Multiply  60  by  ^\. 

3.  Multiply  76  by  J |. 

Ans.  15, 
Ans.  16, 
Ans.  26, 

FRACTIONS.  127 


4.  Multiply  169  by  J|. 

5.  Multiply  5  by  y\. 

6.  Multiply  320  by  /g. 

7.  Multiply  480  by  j|. 

8.  Multiply  75  by  f . 

9.  Multiply  87  by  ^f . 

10.  Multiply  147  by  /g. 

11.  Multiply  1728  by  J, 


5      7       ^5  9_      1  ; 


^ns. 

156. 

J.?25 

^2f 

J-TiS. 

180. 

Ans. 

440. 

Ans. 

64f. 

Ans 

■.39. 

Ans. 

lOi. 

,  and 

U' 

Last  Ans.  1692. 

156.  To  multiply  two  or  more  simple  fractions  to- 
gether, 

Place  the  product  of  the  numerators  over  that  of  the 
denominators. 

Remark  1. — The  rule  applies  also  to  tlie  reduction  of  compound 
fractions  to  simple  ones, 

EXAMPLES. 

1.    Multiply  I  by  | ;  that  is,  find  the  value  of  g  of  |. 


OPERATION. 

fX?-/5^^^^- 

2. 

Multiply  i  by  |. 

Ans.  |, 

3. 

Multiply  1  by  f . 

,         Ans.  IJ, 

4. 

Multiply  f  by  ji. 

^ns.  if, 

5. 

Multiply  ^  by  II 

Ans.  If?, 

6. 

Multiply  f  by  |. 

^^s.  If. 

7. 

Multiply  ,^3  by  y. 

Ans.  .v.. 

8. 

• 

Multiply  1  by  f 

J.WS.  if. 

9. 

Multiply  II  by  If. 

^r^s.  f  if 

10. 

Multiply  1  by  |  of  f. 

Ans.  3^0. 

11. 

Multiply  ,3,  of  1  by  f . 

^?js.  -igl, 

12. 

Multiply  i  of  1  by  ^'  of  |>. 

Ans.  4||. 

128  FRACTIONS. 

13.  Multiply  V  of  i  ^J  A-  ^^^'  ioi- 

14.  Multiply  f  of  I  of  /j  by  V._(yi(ie  147,  Ex.  9.) 

Remark  2. — All  the  factors  common  to  the  numerators  and  de- 
nominators should  be  canceled  before  multiplying. 

15.  Multiply  f  of  V  of  V  ^J  t\- 

OPERATION. 

f     X    V    X      V      X    i\.     Given  fractions. 

^     ^^  ^X^  ^  ^X2X2  ^  ^X^~^' 

16.  Multiply  f  I  of  fl  of  J  by  /g.  ^tzs.  f. 
.  17.   Slultiply  f  §  of  1/  of  V  by  J^.             J.m.  8. 

■    _18.   Find  tlje  value  of  fX  VX^j^.         A71S.  511 1. 

19.   Findthe^valueof  JfXfJXfXlXV- 
'  A71S.  58f  §. 

V     »  20.   Find  the  value  of  ;|gXVXifgX|X-|Xf. 
.     \       /.    '  ^^«-  24j3y. 

#   '21,  Find  the  value  of  ^X  V-X|fgX4Xi>^f 

*   %   >-^  ■'  .  A71S.  s^di. 

♦  ,       ^2.  -^Firid  the  value  of  %^  X  3^-  >^10.  Ans^20. ' 

GENERAL    RULE. V  '^  5f 

157.    To  multiply  fractional  numbers,  ^.  *^  • ' 

(1.)  In  compound  fractions  consider  the  word  of  as  a 
sign  of  multiplication. 

(2.)  Reduce  complex  fractions,  integral  and  mixed 
numbers,  to  simple  fractions. 

(3.)  Cancel  all  the  factors  common  to  the  numerators 
and  denominators. 

(4.)   Multiply  the  remaining  factors  of  the  numerators 


FRACTIOXS.  129 

iog ether,  and  also  those  of  the  deiiominators,  forming  a 
simple  fraction  of  the  products. — (Vide  153,  Rem.  1.) 

Remark. — The  best  mode  of  canceling  common  factors  will  be 
learned  by  practice.  It  is  hardly  ever  necessary  to  resolve  each 
term  into  lis  prime  factors. 

EXAMPLES. 

1.  Multiply  2^X61x31X1^3X2x1,  forming  a  simple 
inber. 

OPERATION. 


fraction  or  integral  nuinber, 


^         ^          ^        73        1        //  1 
Draw  a  line  across  4  and  32,  writing  8  in  place  of  32. 

2.  Multiply  2 J  by  2h  #     Ans.  61. 

3.  Multiply  61  by  61.           ^      ^  ^  -Ans.  Z^-^^. 

4.  Multiply  31  by  H.  r4Nl  Ans.  l^. 

5.  Multiply  2i  by  ^.   ^      ^  ^    Ai^.  1^\. 

6.  Jiultiply  5^  by:5i/'  Ans.  30i. 
"7.'  fli^tiply  XQV^y  16^.  Ans.  272i. 

8.  Multiply  4i;  byv4i.^i%  Ans.  20i. 

9.  Multiply  7i  by  1'^ig  >  Ans.  h^. 

10.  Multiply  ^  by^3j  t     ^  .Aws.  8f . 

11.  Multiply  i\  hj!%.  Ans.  1. 
,12.  Multiply  8^  by  3  J^.  Ans.  2B. 

13.  Multiply  91  by  2^\.  Ans.  18J. 

JL4.  Multiply  3}  by  3^^.  '       Ans.  9 J. 

15.  Multiply  21  by  4^6.  Ans.  8f. 

16.  Find  the  value  of  I4x||-X^. 

OPERATION. 

fifXTViXlg.     (Vide  154,  Rem.) 

mX4        5X10        5X3  _  ^  3  . 

113X5^3X47  ^5X2~-^^-  ^'''' 


X)  niACTIONS. 

J 

17.  Find  the  value  of  ?X^x4  Am.  |. 

18.  Find  the  value  of  ^X^fX^f  •         Am.  1/A. 

19.  Find  the  value  of  ^X#,X^X207. 

Am.  16|. 

20.  Find  the  value  of  4  X  8  ^^  X  f  X  V  X  24 J. 

J.72S.  614^. 

21.  Find  the  value  of  |^x||xgx^. 

Am.  If. 

22.  Find  the  value  of  f  X^^X^xp'.     Am.  ^%^^%. 

23.  Find  the   value   of  V/XyX— ^xSx^  of  ^ 

"  465       3^      2^  ' 

X5i.  ^^s.  132. 

24.  Find  the  value  of  1  §  x  3^0,0  x  §  X  |  X  ^s""  X 15. 

Ans.  120. 

25.  Find  the  value  of  T\X|X|Xy.         Am.  1^. 

26.. Find  the  value  of  IxO/^Xf X4|X8|. 

Am.  240|. 

27.  Find  the  value  of  f  gX  }iX  V0VXI8. 

Ans.  9. 

28.  Find  the  value  of  5iX5i  XS^.         Ans.  166|. 

29.  Find  the  value  of  3^  x^lxS^  X3|. 

A71S.  150  Jg 

DIVISION   OF   FRACTIONS. 

158.  To  divide  a  simple  fraction  by  an  integral 
number, 

(1.)  Divide  the  numerator  by  the  integral  number,  if  it 
is  exactly  divisible,  and  place  the  quotient  over  the  de- 
nominator; otherwise, 


FRACTIONS.  131 

(2.)  Multiply  the  denominator  hy  the  integral  nitmher, 
and  place  the  product  under  the  numerator.  —  (Vide 
145,  III.) 

EXERCISES. 


5.  Divide  |  by  5.         Am.  -^. 

6.  Divide  y  by  4.     Ans.  1^^. 

7.  Divide  %^  by  13.     Ans.  If. 

8.  Divide  f|  by  19.       Ans.  ^. 


2.  Divide  ^%  by  18.     Ans.  ^^ 

3.  Divide  y  by  36.     Ans.  ^\. 

4.  Divide  ^  by  51.  Ans.  ji-^. 

9.  Divide  JiJ§  by  11,  12,  13,  14,  15, 16, 17,  and  18. 

A.71S.    2  2  0'  2  4  05   ^^^' 

10.  Divide  ^^g^  ]by  17;  s^o^e  by  18;  ^^^^  by  19,  and 
36ihv1Q  '  An9   §'    1^-    g    and  J  9 

150.  To  divide  a  mixed  number  by  an  integral 
number. 

Reduce  the  mixed  number  to  an  improper  fraction,  and 
then  divide  as  in  158. 

EXERCISES. 

1.  Divide  2J  by  5.— (Vide  Ex.  5,  158.) 

2.  Divide  7 J  by  8.  Ans.'\l. 

3.  Divide  9|  by  12.  Ans.  f . 

4.  Divide  Ij  by  2.  Jltis.  f. 

5.  Divide  5f  by  4.  J.?is.  l/^- 

6.  Divide  2-i  by  5.  A71S.  /g. 

7.  Divide  If  by  36.  Ans.  ^^. 

8.  Divide  41  by  6.  '  Ans.  ^^. 

9.  Divide  ^  by  11.  J.7is.  f . 
10.  Divide  5j  by  13.  A71S.  §. 

Remark  1. — If  the  dividend  is  larger  than  the  divisor,  its  inte- 
gral part  may  be  first  divided  for  the  integral  part  of  the  answer; 
then  divide  the  remainder  united  ^  the  fractional  part  of  the  divi- 
dend for  the  fractional  part  of  the  answer. 


132  FRACTIONS. 

11.   Divide  27-1  by  5;  71^  by  8;  and  153f  by  12. 


OPERATIONS. 

(1.)                       (2.) 

(3.) 

5)271                   8)711 

12)153f 

5/5(yideEx.l)  8li(yideEx.i 

2.)     12|(VideEx.3.) 

12.   Divide  54i|  by  2. 

Ans.  27/^. 

13.  Divide  117i  by  4. 

Ans.  Ex.  5, 154. 

14.  Divide  1831  by  15. 

Ans.  Ex.  6,  154. 

15.   Divide  150/g  by  11. 

.    Ans.lS}^. 

16.   Divide  1641  by  12. 

Ans.  im. 

17.  Divide  471  by  5. 

A71S.  9/5. 

18.   Divide  441  by  6. 

Ans.  7|. 

19.   Divide  100  by  3. 

Ans.  331. 

20.   Divide  1274J  by  11, 

12,1^ 

5,  14,  15,  16,  17,  18, 

and  19. 

Ans. 

115||,  106^4,  etc. 

Remark  2. — If  the  divisor  is  a  composite  number,  it  is  usually 
best  to  divide  first  by  one  of  the  component  parts,  and  the  resulting 
quotient  by  another  part,  and  so  on  till  all  the  component  parts  are 
used. 

21.   Divide  4567f  by  25,  32  and  51. 


OPEKATIONS. 

(1-) 

(2.) 

(3.)- 

5)4567| 

4)4567| 

3)4567f 

5)  913>-J 

Ans. 

8)1141}i 
142  i»A 

Ans. 

17)1522/^ 

182tV0 

89i-Jf  Ans. 

22.   Divide  34027f  by  36,  42,  65,  56,  72,  and  49. 
Ans.  945,^6,  810, Vg,  ^23,^6,  etc. 


FRACTIONS.  133 

23.  Divide  72431  i  by  15,  21,  28,  and  35. 

Ans,  4828f  3,  3449 Z^,  etc. 

160.  To  divide  an  integral  number  by  a  simple 
fraction, 

Multiply  the  whole  niimher  hy  the  denominator  of  the 
fraction,  and  divide  the  product  by  the  numerator ;  or, 

Invert  the  divisor,  and  then  multiply  t?s  m  155,  (2.) — 
(\^ide  138,  Rem.  2.) 


EXERCISES. 

1. 

Divide  15  by  f. 

Ans.  20. 

2. 

Divide  16  by  j\. 

Am 

f.  Ex.  2,  155. 

3. 

Divide  26  by  ^f. 

Ans.  Ex.  3. 

4. 

Divide  156  by  a|. 

Ans.  Ex.  4. 

5. 

Divide  3  by  5. 

^ns.  2i. 

6. 

Divide  49  by  J. 

Ans.  28. 

7. 

Divide  57  by  '/. 

J.71S.  24. 

8. 

Divide  32  by  V- 

J.??s.  114. 

9. 

Divide  45  by  J,  |, 

f. 

h 

h\h 

1  6 
16? 

V,  and  ^%. 

Ans.  90,  67J,  60,  etc. 

10.  Divide  4290  by  f  f ,  VoS  ^^^l  f  |. 

^ns.  2015,  etc.,  (vide  123,  Ex.  16.) 

11.  Divide  28560  by  ^p,  ^f  ^  YrS  and  V/- 

^ns.  525>  840,  1540,  2184. 
161.    To  divide  one  simple  fraction  by  another, 
Invert  the  divisor,  and  then  place  the  product  of  the 
numerators  over  that  of  the  denominators. 


EXAMPLES. 

1.  Divide  f  by  |. 
2    Divide  |  by  |. 

Ans.  \ 
Ans. 

134 


14 

FRACTIONS. 

3. 

Divide  |  J  by  |. 

Ans.  |. 

4. 

Divide  §1  by  J  J. 

Ans.  If. 

5. 

Divide  f^_o.^^  II . 

JLtzs.  ^^. 

6. 

Divide  3y,  by  i|. 

A71S.  |. 

7. 

Divide  Jgf  by  |f. 

^Tis.  /r- 

8. 

Divide  |  by  f . 

Ans.  Jf . 

9. 

Divide  -/  by  |. 

^/^s.  10. 

10. 

Divide  J|  by  V- 

J.n.s.  1^\. 

11. 

Divide  If?  by  ]J. 

Ans.  ;|. 

12. 

Divide  |f  |  by  If. 

^7ZS.  if. 

13. 

Divide  J  by  f. 

Ans. 

Ex.  14,  154. 

14., 

Divide  ^%  by  j%. 

Ans. 

Ex.  13,  154. 

GENERAL    RULE. 

162.    To  divide  fractional  numbers, 

(1.)  In  compound  fractions  consider  the  tvord  of  as  a 
sign  of  multiplication. 

(2.)  Reduce  complex  fractions^  integral  and  mixed 
numbers,  to  simple  fractions. 

(3.)  Invert  each  of  the  reduced  or  given  simple  frac- 
tions considered  as  divisors, 

(4.)  Cancel  all  the  factors  common  to  the  numerators 
and  denominators  of  the  simple  fractions. 

(5.)  Multiply  the  remaining  factors  of  the  numerators 
together,  and  also  those  of  the  denominators,  forming  a 
simple  fraction  of  the  products. — (Vide  153,  Rem.  1.) 

EXAMPLES. 

1.   Divide  4  of  §  of  5i  by  if  of  48. 

OPERATION. 

4X|XVXifX^xV5=/6  ^ns. 


FRACTIONS.  135 

2.  Divide  f  of  |  by  J  of  2^.  Ans.  Jf. 

3.  Divide  39-jig  by  6\.  Ans.  Gj. 

4.  Divide  272}  by  16\,  Am.  16j. 

5.  Divide  3061  by  17 J.  Ans.  17^. 

6.  Divide  f  of  J/  by  ^\  Ans.  1. 
•  7.   Divide  1419^X11X^5  by  ^.              ^?is.  2f. 

8.  Divide  i|x^  by  ^,Xl8f .  Ans.  f  . 

9.  Divide  4iX^"  by  %^X~.  Ans.  I. 

10.  Find  the  value  of  ^X-f  divided  by  ^XlOj. 

Ans.  ii^. 

11.  Find  the  value. of  |X^  divided  by  ?X^X^| 
X^-lf .  ^^^        ^    •  ^.;.  i-.   ' 

12.  Find  the  value  of  ilX^Xyi   divided  by  ^X 

167  ^  6f 

13.  Find  the  value  of  2i-  divided  by  5i. 

Ans.  Ex.  14,  154. 

14.  Find  the  value  of  ?±-^.         Ans.  Ex.  14,  154. 

15.  Find  the  value  of  ^4-  -^^s-  lA- 

16.  Reduce  |  to  a  fraction  whose  denominator  shall 
be  4.— (Vide  148.)  Ans.  ^. 

17.  Reduce  J|  to  a  fraction  whose  denominator  shall 
be  2J.— O^ide  154,  Ex.  12.)  Ans.  ^. 

3 

18.  Reduce  J  J  to  a  fraction  whose  denominator  shall 
be  5i.  Ans.  rr. 


X^.  ^ns.  5if. 


H 


136  FRACTIONS. 

REDUCTION  OF  COMMON  FRACTIONS  TO  DECIMAL 
FRACTIONS. 

163.  To  reduce  a  simple  fraction  to  a  decimal 
fraction, 

Divide  the  numerator  hy  the  denominator. — (Vide  92, 
Rem.  3,  and  145.) 

EXAMPLES. 

1.  Reduce  J,  |,  |,  and  ]|,  to  decimal  fractions. 

OPERATIONS. 

(1.)  (2.)  (3.)  (4.) 

2)1.0      .       4)3.00  8)5.000  16)13.0000 

.5  Ans.         .75  Ans.        .625  Ans.  .8125  Ans. 

2.  Reduce  ^-q,  ^l^,  iJif?  ^^^  IS  *^  decimal  frac- 
tions.— (Vide  94  and  95,  Rem.  3.) 

Ans.  .1,  .03,  1.728,  and  .925. 

3.  Reduce  -||,  H,  ff,  ^^-3%,  ^i^,  i^%,  and  if  to  deci- 
mal  fractions. 

Ans.  .4,  .85,  .5,  .025,  .128,  .096,  and  .36. 

4.  Reduce  ^y^,  J/^,  ,-%,  /„  hi,  |f,  and  ,-Ji^  to 
decimal  fractions. 

Ans.  .06640625,  .08,  .4,  .25,  .48,  .0765625,  .006875. 

5.  Reduce  ^fS  n^,  f§,  i§,  0^,  ||,  and  W  to 
decimal  fractions. 

Ans.  30.25,  35.2,  2.0625, 1.2,  4.09375,1.171875,  5.08. 

6Rp(lnPP    3      26        7       210        80       24      nrifl    5  1    f^   /Ippi'mnl 

fractions.       Ans.  1.5,  2.5  .28,  8.75,  .128,  1.6,  and  .75. 

7.  Reduce  1^,  3j,  81,  6/^,  30f,  35f,  and  7S  to 
mixed  decimals. 

^?zs  1.5,  3.25,  8.2,  6.7,  30.75,  35.6,  and  7.625. 


FRACTIONS.  137 

8.  What  is  the  value  of  ^3^,  $4|,  $7f ,  $Sl,  and  ^6\  |? 

Ans.  $3.50,  §4.625,  |7.75,  §8.20,  and  §6.81i. 

Remark. — If  the  denominator  of  a  fraction  in  its  lowest  terms 
contains  a  prime  factor  other  than  2  or  5,  the  value  of  the  fraction 
can  not  be  exactly  expressed  by  a  decimal.  The  exact  value  may, 
however,  be  preserved  by  placing  the  excess  of  fractional  units  to 
the  right  of  the  quotient,  stopping  the  division  at  pleasure. 

9.  Reduce  ^,  |,  f.j,  f,  |,  and  y\  ^^  mixed  decimals. 

Ans.  .331,  .16§,  .416f ,  .66|,  .83J,  and  .583i. 

10.  Reduce  |,  J^,  fj,  |,  f  J,  |f,  and  ||  to  mixed 
decimals. 

Ans.  .4281,  .06|,  .846/3,  .33^,  1.16f ,  .66|,  and  1.66f . 

11.  What  is  the  value  of  $7^,  |4|,  |6|,  $8J,  and 
§12  J^? 

A71S.  §7.33J,  |4.83i,  |6.66|,  ?8.773,  and  |12.06|. 

164.  To  reduce  a  decimal  fraction  to  a  common 
fraction  in  its  lowest  terms, 

(1.)  For  the  numerator  of  the  fraction,  write  the 
figures  composing  the  given  number. 

(2.)  For  the  denominator,  write  1,  with  as  many 
cij)he7's  annexed  as  there  are  decimal  places  in  the  given 
number. 

(3.)   Reduce  the  resulting  fraction  to  its  loivcst  terms. 

EXAMPLES. 

1.   Reduce  .5,  .75,  .625,  and  .8125  to  simple  fractions. 

OPEKATIONS. 

(1.)  (2.)  (3.)  (4.) 

,»,=jAns.  j\%=\Am.  {ii-,=lAm.   -^^yii,^\lArw. 
12 


138  FRACTIONS. 

2.  Reduce  .06640625  and  30.25  to  simple  fractions. 

OPERATIONS. 

(1.)  (2.)  or  (2.) 

3.  Reduce  .25,  .85,  .55,  .125,  .135,  and  .325  to  simple 
fractions.  Aiis.  i,  JJ,  J  i,  -J,  i^\,  and  i-§. 

4.  Reduce  .025,  .0085,  .9375,  .0008,  and  .16  to 
simple  fractions.        Ans.  J^,  ^ij^,  i|,  ^  J^o'  and  5*^. 

5.  Reduce  .34375,  .1328125,  and  .203125  to  simple 
fractions.  Ans.  J  J,  y^g,  and  ||. 

6.  Reduce  $3.50,  $4,625,  $7.75,  §8.20,  and  $3.40  to 
dollars.  Ans.  $3j,  $4|,  $7f ,  $8|,  and  $3f. 

Remark. — "When  there  is  an  irreducible  fraction  at  the  end  of 
the  decimal, 

(1.)  Consider  the  given  number  as  integral^  and  reduce  it  to  an  im- 
proper fraction. 

(2.)  Annex  to  the  denominator  as  many  ciphers  as  there  are  decimal 
p)laces  in  the  given  number. 

(3.)  Reduce  the  resulting  fraction  to  its  lowest  terms. 

7.  Reduce  .428|,  .066|,  and  .83j  to  simple  fractions. 

OPERATIONS. 

(1.)  (2.)  (3.) 

.428f  .066|  .83i 

3000 S-d/Mo  200  1        J*iQ  2  50 5     A'i'}^ 

8.  Reduce  .47J,  4.7|,  47.33i,  .047^,  and  .43J  to 
simple  or  mixed  fractions. 

^ns.  ^5^5,  4j-i,  47^,  tJJ^,  and  J^. 

9.  Reduce  .473j,  .33^,  .45|,  .125^,  and  .25|  to  simple 
fractions.  Ans.  -jVo>  irs^  ft?  /^^j  ^^^  «o- 


FRACTIONS.  189 

10.  Reduce    §7.33|,    |4.83|,   |7.25J,  |0.754f,  and 
|4.50f  to  dollars. 

Ans.  $7J,  Uh  ^n,  Psih  and  |4f|. 

11.  Add  together  $240,172,  §120.75f ,  and  §255.136f . 

Ans.  $616,066^. 

12.  Add  together  $5.87|,  $3,187^,  and  |2|. 

^  ^718.  $11,687^. 

13.  Add  together  |,  .066f ,  and  .8]. 


OPERATION. 

1=     .4281 

.066| 

M=     .833A 


1.328f=:lf g  Ans. 


14.   Add  together  f ,  .18|,  and  7-|. 


Ans.  8.7652|=8f  |i. 

15.  From  $25.41  take  $17|.  Ans.  $7.66. 

16.  From  $28,026  take  $19.15|.  Ans.  $8.872f . 

17.  From  $12.25  take  $8i.  J^ns.  $3.75. 

18.  From  $5|  take  $4.25.  Ans.  $1,08 «. 
19.*  From  $6i  take  $5.25.  Aiis.  $1.00. 
20.    From  $7f  take  $1|.         "  ^?^s.  $6.58'. 

165.   Problems  Involving  Preceding  Principles. 

1.  If  a  horse  consume  ^  a  bushel  of  oats  in  one  day, 
I  of  a  bushel  in  another,  |  of  a  bushel  in  another,  and 
f  of  a  bushel  in  another,  how  many  bushels  are  con- 
sumed in  the  four  days? — (Vide  151,  Ex.  1.) 

Ans.  2|g  bushels- 


140  FKACTIONS. 

2.  If  I  buy  4  of  ^  yard  of  ribbon  at  one  store,  y\  at 
another,  o^-  ^^  another,  and  -f^  at  another,  hoAV  many 
yards  have  been  purchased? — (Vide  151,  Ex.  13.) 

Ans.  if  J  yards. 

3.  A  man  bought  two  pieces  of  cloth,  one  containing 
13f  yards,  and  the  other  14|  yards.  How  many  yards 
in  both  pieces  ?— (Vide  151,  Ex.  21.) 

Ans.  28^  yards. 

4.  In  one  pile  of  wood  I  have  4 J  cords;  in  another 
3J  cords;  in  another  4i;  and  in  another  6^  cords. 
How  many  cords  of  wood  in  the  four  piles? — (Ex.  25) 

Ans.  18j2§  cords. 

5.  If  I  make  purchases  to  the  amount  of  'fl7|-|  at 
one  time,  and  §16||  at  another,  how  many  dollars  have 
I  expended  in  all?— (Ex.  24.)  Ans.  $34i. 

6.  If  I  travel  47 f  miles  in  one  day,  33  ig  in  another, 
and  19  in  another,  how  many  miles  have  I  traveled  in 
all?  Ans.  100  miles. 

7.  Add  together  §4.25i,  §3.37 J,  §6.753,  and  §7.52  j^,. 

Ans.  §21.91.  ' 

8.  Add  together  §240.17^,  §120.75|,  and  §255.13f . 

A9is.  §616.06J2. 

9.  Add  together  §16.254,  §40.20|,  §13.60 J,  and 
§24.035.  Ans.  §94.094 J. 

10.  What  is  the  difference  between  §  of  a  dollar  and 
2'j-  of  a  dollar?— (Vide  152,  Ex.  1.)         A7is.  §0.16|. 

11.  What  is  the  difference  between  §3.00  and  -|  of  a 
dollar  ?  Ans.  §2.25. 

12.  From  a  cask  of  wine  containing  8  gallons,  2J 
gallons  were  drawn.  What  quantity  remained  in  the 
cask?— (Vide  152,  Ex.  21.)  "^Ans.  5j  gallons. 


FRACTIONS.  141 

13.  If  I  purchase  flour  at  3 J  dollars  per  barrel,  and 
sell  it  for  5|  dollars,  ^'hat  doJ^  gain?       Ayis.  |1.83j. 

14.  A  farmer  sold  55  y^j-  ajp^  from  a  farm  of  100  acres. 
IIow  many  acre^did  lie  s^  own?  ^  A71S.  4:4: f^  acres. 

15.  If  I  buy  a'p'iece  of  land  for  |1 03.33 J,  and  sell 
the  same  for  $106.66 f,  Avhat  do  I  gain  ?     Ans.  $3.33  J. 

16.  A  merchant  bought  a  piece  of  cloth  containing 
123|  yards,  and  from  it  sold  45j'^(j  yards.  How  many 
yards  remained?  Ans.  78 /^  yards. 

17.  A  railroad  train  has  13 1  hours  in  which  to  run 
550  miles.  Having  run  lOj  hours,  the  conductor  finds 
that  only  41 6|  miles  have  been  made.  What  distance 
is  yet  to  be  run,  and  in  what  time? 

Ans.  133  J  miles  in  2|  hours. 

18.  From  1000  yards  of  cloth  I  sold  at  one  time 
479j'^g  yards,  and  at  another  275 1.  How  many  yards 
have  I  still  on  hand?  A71S.  245 /g. 

19.  A  merchant  bought  at  one  time  234|  yards  of 
cloth,  at  another  time  753y'^Q  yards,  and  sold  of  the  two 
lots  843^^  yards.     How  much  cloth  has  he  yet  on  hand? 

Ans.  145 i  yards. 

20.  I  bought  30  cords  of  wood  for  $105 J-,  and  sold 
17  cords  of  the  wood  for  $65,25.  If  I  sell  the  remain- 
ing 13  cords  for  $45^,  how  much  do  I  gain  in  the  trans- 
actions? Ans.  $5,585. 

21.  The  sum  of  two  numbers  is  4b^ ;  one  of  them  is 
23/2-.     What  is  the  other  ?— (Vide  151,  Ex.  23.) 

Ans.  21|. 

22.  The  difference  between  two  numbers  is  J,  and 
the  less  is  4|.     What  is  the  greater?  Ans.  4|. 

23.  The  difference  between  two  numbers  is  13?,  and 


142  FRACTIONS. 

the  greater  is  28 f|.      What  is  the  less? — (Vide  151, 
Ex.  20,  (3.)  ^  .^        A71S.  14f . 

24.  The  difference  betA^kn  two  nuiflbers  is  184i,  and 
the  greater  is  509^.  What  il  the  sumVf^the  two  num- 
bers?—(Vide  125"rte*x.  9.)        .       "  •    '  Ans.  8343^. 

25.  The  sum  of  two  numbers  is  34^3,  and  one  of 
them  is  16||.     What  is  the  other?  Ans.  ITJ. 

26.  The  sum  of  four  numbers  is  89ff.  Three  of  the 
numbers  are  65,  7^^,  and  SjL.     What  is  the  fourth? 

Ans.  9^. 

27.  A  farmer,  who  had  wheat  worth  |4325.75,  sold  at 
one  time  120  bushels  for  $135  J ;  at  another  time  45J 

"imshels  for  §70 1 ;  and  at  another  time  87|  bushels  for 
/,  »;  ii60.  What  is  the  value  of  the  4000  bushels  he  still 
.         finds  he  has  on  hand?  Ans.  $4019.58 J. 

'^^  *^  4  lOO,    To  find  the  cost  of  a  number  of  things,  when 
jy^  1!||e  cost  of  one  i^  given, 

**^^  iKidtipm  the  cost  of  one  hy  the  .numher  of  tilings.     The 
t*  ^  plbduct  ^vi'il  be  the  cost  of  the  w^hole. — (Vide  82.) 
^  •*.   ^*28f^If*lt  yard  of  cloth  cost  |  of  a  dollar,  what  will 
*'*-4^;^ards  cost?     (Vide  153,  Ex.  1.)     15  yards?   20? 
40?  J.ns.  $6.00,  etc. 

29.  If  a  pound  of  butter  cost  |  of  a  dollar,  what 
will  6  pounds  cost?  (Vide  153,  Ex.  3.)  8  pounds? 
12?  20?  Alls.  $2.25,  etc. 

30.  If  a  pound  of  raisins  cost  ^^  of  a  dollar,  what 
will  5  pounds  cost?  (Vide  153,  Ex.  5.)  10  pounds? 
12?  15?  20?  Ans.  $2.33 J,  etc. 

31.  At  J§  of  a  dollar  a  pound,  what  will  15  pounds 
of  beef  cost?    (Vide  Ex.  7.)    20  pounds?  30?  45?  60? 

Ans.  $3.25,  etc. 


FRACTIONS.  143 

32.  At  J I  of  a  dollar  a  bushel,  what  will  20  bushels 
of  apples  cost?  (Vide  Ex.  11.)  25  bushels?  30?  35? 
40?  Ans.  §13.60,  etc. 

33.  If  a  ton  of  hay  cost  |27i^3,  what  will  be  the 
cost  of  2  tons?  (Vide  154,  Ex.  4.)  3  tons?  4?  5? 
12?  Ans.  154.76; I,  etc. 

34.  If  one  coat  cost  13  J  |  dollars,  what  will  be  the 
cost  of  11  coats?  12?  13?  14?  15? 

Ans.  $150,561,  etc. 

35.  If  a  ton  of  hay  cost  $20,  what  will  |  of  a  ton 
cost?     (Vide  155,  Ex.'l.)     |  of  a  ton?  ^^'i  -/^? 

'    '\       ^  ^    ^^  Ans.  $15.00,  etc. 

36.  If  a  bale  of  cott<)^  cost  ,^6p,  what  will  /g  of  a 
bale  cost ?     (Vide  Ex.  2.) "  ^\'?  .^% ?  'i § ?  |J ? 

Ans.  $16.00,  etc. 

37.  When  flour  is  woYtJi  $5  a  barrel,  what  is  the 
value  of  j^5  of  a  barrel?   o%?  -/^  ?  J§? 

Ans.  $2,331,  etc. 

38.  If  a  flock  of  sheep  is  wbrth  $1728,  what  are  | 
of  it  worth"?   ^'^  l'^  A?  -9  ?  13?  4  7  7' 

Ans.  $1382.40,  etc. 

39.  If  an  acre  of  land  is  worth  $128,  what  are  -|  of 
it  worth?  ^'^  -%.*?  -K'^  -5- *?  Si?   is? 

Ans.  $112.00,  etc. 

40.  If  a  yard  of  cloth  cost  §  of  a  dollar,  what  are  | 
of  a  yard  worth?— (Vide  156,  Ex.  1.)      Ans.  $0.174 . 

41.  If  a  yard  of  cloth  cost  f  of  a  dollar,  what  are  f 
of  a  yard  worth?  Ans.  $0,174. 

42.  If  a  pound  of  tea  cost  \  of  a  dollar,  what  cost 
f  of  a  pound?  (Vide  156,  Ex.~2.)  |  of  a  pound?  4? 
i  ?  ^  ?  A71S.  37i  cents,  etc. 


144  FRACTIONS. 

43.  If  silk  is  worth  ||  of  a  dollar  a  yard,  what  are  ^\ 
of  a  yard  worth?  Ji?  ^\  ?   ff  ?         Ans.  ?0.112|0f. 

44.  What  will  -^\  of  a  pound  of  tea  cost  at  J  of 
a  dollar  a  pound?  f?  |?  |?  J  of  ^  of  a  dollar  a 
pound?    •  Last  Ans.  $0.07 o\. 

45.  At  2 J  cents  each,  what  will  be  the  cost  of  2h 
apples?     (Vide  157,  Ex.  2.)    Sj?   4j?   4i?  12i  ?  12i? 

Ans.  6\  cents,  etc. 

46.  At  16^  dollars  an  acre,  what  will  16^  acres  of 
land  cost?  25|?  SO-J?  401?  75|?  JLtis.  |272.25. 

167.  To  find  the  cost  of  one  thing  when  the  number 
of  things  and  the  cost  of  the  whole  are  given, 

Divide  the  cost  of  the  tvhole  hy  the  number  of  things. 
The  quotient  will  be  the  cost  of  one. — (Vide  96.) 

47.  If  2  yards  of  cloth  cost  f  of  a  dollar,  what  will 
one  yard  cost?— (Vide  158.)  Ayis.  $0.16|. 

48.  If  18  yards  of  ribbon  cost  ^^  of  a  dollar,  what 
will  one  yard  cost?  Ans.  2\  cents. 

49.  If  5  sheep  cost  27j  dollars,  what  will  be  the 
price  of  one  sheep? — (Vide  159,  Ex.  11,  (1.) 

Ans.  |5.46f . 

50.  If  8  yards  of  broadcloth  are  worth  71 J  dollars, 
what  is  one  yard  worth  ?  Ans.  -$8.9 If 

51.  If  12  acres  of  land  cost  153|  dollars,  what  will 
one  acre  cost?  Ans.  §12.80. 

52.  If  6  pounds  of  butter  cost  2  J  dollars,  what  will 
one  pound  cost?  Ans.  37^  cents. 

53.  If  9  bushels  of  wheat  cost  10^-  dollars,  what  will 
one  bushel  cost?  Ans.  $1.16|. 

54.  If  4  acres  of  land  cost  fll7|.  Avliat  will  one  acre 
of  the  same  land  cost?  Ans.  §29.45. 


FRACTIOXS.  l-i5 

55.  I  have  four  small  farms  containing,  respectively, 
11,  12,  13,  and  14  acres,  and  I  value  each  farm  at 
|1274j.  What  is  the  value  per  acre  of  each  farm  ? 
What  is  the  average  price  per  acre  ? 

Ans.  $115.8G^*j-,  $10G.208i-,  $98.038yfij,  $91,035^,  $101.96. 

56.  If  I  of  a  ton  of  hay  cost  §15,  what  is  the  price 
per  ton  ?— (Vide  160,  Ex.  1.)  Ans.  §20. 

57.  If  y'^W  of  a  man's  salary  per  month  amounts  to 
§84,  what  is  his  salary  per  ^ear?  Ans.  §1728. 

58.  A  gentleman  divided  j\  of  a  lot  of  marbles 
among  4  boys,  giving  each  boy  21  marbles.  How  many 
marbles  would  each  boy  have  received  if  the  whole  lot 
had  been  divided  ?  Ans.  36  marbles. 

59.  If  J  of  a  bale  of  cotton  are  worth  §48,  what  is 
one  bale  worth?  Aois.  §54.85|. 

167^.    To  find  the  number  of  things  when  the  cost  .• 
of  one  is  given, 

Divide  the  cost  of  all  hy  the  cost  of  one  thing.  The 
quotient  will  be  the  number  of  things? — (Vide 96.) 

60.  At  12f  dollars  an  acre,  how  many  acres  of  land 
can  be  bought  for  153 1  dollars?  Ans.  12  acres. 

61.  At  I  of  a  dollar  a  pound,  how  many  pounds  of 
butter  can  be  bought  for  2  J-  dollars.      Ans.  6  pounds. 

62.  If  cherries  are  worth  7|  cents  a  quart,  how  many 
quarts  can  be  bought  for  1^^^  dollars? 

Ans.  16  quarts. 

63.  How  many  sheep  can  be  bought  for  27-|  dollars, 
if  the  average  price  is  §5.46f .  Ans.  5  sheep. 

64.  At  2 J-  dollars  a  yard,  how  many  yards  of  cloth 
can  be  bought  for  ^^-^  dollars  ?  Ans.  3  J  yards. 

13 


146  FKACTIONS. 

65.  If  a  man  boards  at  3|  dollars  a  week,  how  long, 
can  he  board  for  188. i  dollars?  Ans.  52  Avecks. 

66.  At  2  dollars  a  yard,  how  many  yards  can  be 
bought  for  5i  dollars?  ^ws.  2|  yards. 

67.  If  I  pay  656J  dollars  for  75  tons  of  coal,  what 
is  the  price  per  ton  ? 

68.  If  a  ton  of  coal  is  worth  8|  dollars,  what  will  bo 
the  cost  of  80  tons  ? 

69.  If  I  pay  §656.25  for  75  tons  of  coal,  what  will 
be  the  cost  of  80  tons  ? 

70.  If  75  tons  of  coal  cost  §656.25,  how  many  tons 
can  be  bought  for  §700  ? 

71.  Ii  80  tons  of  coal  cost  §700,  how  many  tons  can 
be  bought  for  §656.25  ? 

72.  I^  75|  yards  of  cloth  cost  §643.87|,  how  much 
^nust  I  pay  for  36  yards? 

73.  When  §306  are  paid  for  36  yards  of  cloth,  how 
many  yards  can  be  purchased  for  §643  J  ? 

74.  For  36  yards  of  cloth  306  dollars  were  paid. 
How  many  yards  of  the  same  cloth  can  be  purchased 
for  §643.875  ? 

75.  For  75|.jards  of  cloth' I  pay  §643.87^.  How 
many  yards  can  I  get  for  §306  ? 

76.  How  many  coats,  each  containing  1|  yards  of 
cloth,  can  be  made  of  18|  yards? 

77.  What  will  ten  barrels  of  apples  cost  at  1 J  dollars 
per  barrel  ? 

78.  If  10  boxes  of  oranges  cost  18j  dollars,  what  is 
the  price  per  box  ? 

79.  If  10  pounds  of  copper  cost  §18.75,  what  num- 
ber of  pounds  can  be  had  for  §171.25. 


FiiACTlONS.  147 

80.  If  ^p3|P^ifcds  of  copper  can  be  liad  for  171 J 
dollars,  howma^y  pounds  will  18|  dollars  buy? 

81.  A  merchant  sells  |  of  his  ship.  What  part  of  it 
does  he  still  own^  Ans.  J. 

82.  A  merchant  owning  |  of  a  ship  sells  J  of  his 
interest.  What  part  of  the  ship  does  he  still  own  ? — 
(Vide  156,  Ex.  2.)  Ans.  |. 

83.  Having  |  of  an  apple,  I  give  away  half  of  it. 
What  part  of  the  apple  is  now  gone?  Ans.  |. 

84.  Having  |  of  a  ship,  I  sell  half  my  interest  for 
9420  dollars.  What  is  the  whole  ship  worth  at  the  same 
rate? 

85.  If  a  §hip  is  worth  |25120,  what  are  |  of  her 
worth  ?     What  are  |  worth  ? '  g-  ?         Ans.  -|=|21980. 

86.  If  I  of  a  ship  are  worth  $15700,  what  is  the 
value  of  I  ?  ^  ' 

87.  If  J  of  a  bale  of  cotton  are  worth  48  dollars, 
what  is  the  value  of  f  of  a  bale?  Ans.  $41^. 

88.  What  fraction  is  that  to  which  if  i  be  added,  the 
sum  will  be  1  ?  2  ?  3  ?  4  ?  5  ?  Ans.  f ,  |,  etc. 

89.  What  fraction  is  that  to  which  if  J^be  added, 
the  sum  will  be  1  ?  20  ?  45  T 100  ?     Ans.  ~'^,  %%\  etc. 

90.  What  number  is  tllaj  which,  if  it  be  taken  from 
57,  will  leave  a  remainder  of  |  ?  |  ?  ^  J  ?  J|  ? 

A71S.  56|,  56|,  etc. 

91.  What  number  is  that  which,  on  being  added  to 
357/5,  will  make  the  sum  455??  ^72^5?  9570^*5? 

Ans.  983  J,  etc. 

92.  What  fraction  is  that  to  which  if  |-  of  f  be 
added,  the  sum  will  be  1  ?  15  ?  8|  ?  4,^o  ^ 

hsist  Ans.  4i§. 


\ 


148  FIIACTIOXS. 

93.  The  sum  of  two  numbers  is  47-J-,  an{>tlie  differ- 
ence 7l.     What  are  the  numbers  ? — (Vide  125,  Ex.  10.) 

A71S.  21^^  and  19iJ. 

94.  If  a  certain  number  be  divided  by  2 J,  and  the 
quotient  be  multiplied  by  8i,  the  product  diminished  by 
5 4,  the  difference  increased  by  7^,  the  sum  will  be  62 f. 
What  is  the  number?  Ans.  18-|. 

95.  I  have  a  fortieth  interest  in  an  oil  well,  and  am 
willing  to  sell  half  of  it  for  $1340.  What  is  the  value 
of  my  interest,  and  of  the  whole  well,  at  the  same  rate  ? 

Ans.  12680  and  $107200. 

3  T 


96. 

What 

is 

i 

of  360  ? 

1  of  720? 

ii 

of  378?   - 

of  1643? 

Am 

f.  270,  600, 

351 

,  and  636. 

97. 

What 

is 

H 

of 

38?    1 

3  of  331? 
Ans.  ^\8^, 

ifo 
29f 

f  32|? 
,  and  31 1. 

>     98.    270  is  I  of  what  number  ?  600  is  §  of  what  num- 
ber?    351  is  H  of  what  number? 

99.  636  is  Jf  of  what  number?  j\%is  i|  of  what 
number?  29|  is  J  J  of  what  number?  31j  is  ||  of 
what  number? 

100.  J^of  68  is  If  of  what  number?        Ans.  58. 

101.  11  of  341  is  J  J  of  how  many  times  5  ? 

Ans.  781  times  5. 

102.  i|  of  49|  is  If  of 'how  many  thirds  of  18? 

Alts. '9  thirds  of  18. 

103.  On  a  trip  from  New  Orleans  to  New  York,  I 
expend  i  of  my  money,  and  still  have  »^270.  What  did 
the  trip  cost  me  ?  Ans.  §90. 

104.  During  a  storm  a  captain  threw  overboard  ^  of 
his  cargo  of  cotton,  and  still  lias  600  bales  on  board. 
How  many  bales  were  thrown  overboard?     Ans.  120. 


FRACTIONS.  149 

105.  If,  after  reserving  ^^  of  my  wine,  I  sell  351 
gallons,  how  many  gallons  do  I  reserve  ?         Ans.  27. 

106.  Multiply  the  fractions   J   and  |   by  the  least 
common  multiple  of  their  denominators. 

Ans.  3  and  4. 

107.  Multiply   the  fractions   |    and    J  by  the  least 
common  multiple  of  their  denominators. 

Ans.  8  and  9. 

108.  Multiply  the  fractions  y\  and  ^^  Ja^he  least 
common  multiple  of  their  denominators.     ^^B 

Ans.  12  and  5. 

109.  Multiply   35 1   and  15 1   by  the  least   common 
multiple  of  their  denominators.  Ans.  214  and  91. 

110.  Multiply  ^,  -i,  and  |  by  the  least  common  mul- 
tiple of  their  denominators.  Ans.  6,  4,  and  9. 

111.  Multiply  ^,  I,  y^Q,  and  §,  by  the  least  common 
multiple  of  their  denominators. 

Ans.  15,  20,  9,  and  12. 

112.  Multiply  j J-,  JJ,  /g,  and  |  by  the  least  common 
multiple  of  their  denominators. 

Ans.  44,  33,  32,  and  24. 

113.  Which  fraction  is  the  greater,  /^  or  -f^  ? 


Ans.   ^ 


Remark. — Of  two  fractions,  that  wliich  gives  the  gi'eater  product 
on  multiplying  both  by  the  least  common  multiple  of  the  denomi- 
nators is  the  greater. 

114.  Which  fraction  is  the  greater,  J  J  or  J  J  ?  -J-f  or 
i  J  ?  and  by  how  much  ?     A71S.  ]  ^  by  ,1 3 ;  J  J  by  -,-  j^. 


115.    Which  fraction  is  greatest,  J,  /j,  or  f^ 


Alls.  /_. 


116.   If  A  and  B  together  can  do   /^  of  a  piece  of 


150  FRACTIONS. 

work  in  one  day,  and  A  alone  can  do  J  of  it  in  one  day, 
what  part  can  B  do  in  one  day?  Ans.  J. 

117.  If  A  and  B  together  can  do  a  piece  of  work  in 
2|  days,  and  A  alone  can  do  it  in  4  days,  in  what  time 
can  B  alone  do  the  work  ?  Ans.  5  days. 

118.  A  and  B  can  do  ^^  of  a  piece  of  work  in  one 
day;  A  and  C  can  do  j%  pf'the  same  work  in  one  day; 
B  and  C  can  do  J  J  in  one  day.  What  part  of  the  work 
could  al^M^ether^do  in  one  day  ?  What  part  could  A 
alone  d^^Bn'e  day  ?  What  part  could  B  alone  do  in 
one  day 'f^  What  part  could  C  alone  do  in  one  day  ? 

A7is.A\\,  ij;  Ai;  Bi;  Ci. 

119.  A  can  do  a  piece  of  work  in  4  days,  B  in  5 
days,  and  C  in  6  days.  What  part  of  the  work  can  A 
and  B  together  do  in  one  day  ?  What  part  B  and  C 
together  ?  What  part  A  and  C  together  ?  What  part 
can  all  together  do  in  one  day  ?  In  how  many  days  can 
all  together  do  the  work?  Last  Ans.  Iff  days. 

120.  A  cistern  has  three  pipes.  The  first  will  fill  it 
in  2  hours,  the  second  in  3  hours,  the  third  in  4  hours. 
In  what  time  will  the  cistern  be  filled  when  the  three 
pipes  are  running  together  ? 

Ans.  In  ]|  of  an  hour. 

121.  A  cistern  has  3  pipes,  two  at  the  top  and  one 
at  the  bottom.  One  of  the  top  pipes  would  fill  it  in  5 
hours,  the  other  in  6;  but  the  pipe  at  the  bottom 
empties  it  in  8|  hours.  In  what  time  will  the  cistern 
be  filled  when  the  pipes  are  running  together  ? 

Ans.  In  4  hours. 

122.  A  man  and  his  wife  could  drink  a  cask  of  beer 
in  10   iays.     In  the  absence  of  the  man  it  lasted  his 


COMPOUND    NUMBERS.  151 

-wife  30  days.     How  long  Avould  the  man  be  occupied  in 
drinking  it?  A7is.  15  days. 

123.  A,  B,  and  C  could  do  a  piece  of  work  in  .f'^ 
days;  A,  B,  and  D  in  |  days;  A,  C,  and  D  in  J§  days; 
B,  C,  and  D  in  j  §  days.  In  what  time  could  they  all 
do  the  work,  and  in  what  time  could  each  man  do  it 
alone  ? 

Ans.  All  in  Jf  days;  A  in  1;  B  in  2;  C  in  3;  and  D  in  4  days. 


COMPOUND  NUMBERS. 


DEFINITIONS. 

168.  An  ABSTRACT  NUMBER  is  a  number  whose  unit 
has  no  name  other  than  that  given  it  as  a  mere  number. 
Thus,  5,  29,  3i,  are  abstract  numbers. 

169.  A  CONCRETE  or  DENOMINATE  NUMBER  is  a  num- 
ber ivkose  unit  has  a  name  other  than  that  given  it  as  a 
mere  number.  Thus,  5  dollars,  29  feet,  3 J  apples,  are 
concrete  numbers. 

ITO.  A  SIMPLE  NUMBER  IS  a  unit  or  collection  of  units 
of  the  same  kind.  A  simple  number  is  either  abstract 
or  co7icrete.  Thus,  5,  5  dollars;  3^,  3^  apples,  are 
simple  numbers. 

171.  A  COMPOUND  NUMBER  is  a  number  consisting  of 
two  or  more  concrete  numbers  of  different  unit  values,  but 
reducible  to  a  simfle  number.  Thus,  3  feet  4  inches  is 
a  compound  number,  and  equal  to  40  inches,  which  is  a 
simple  number;  8  dollars  5  cents  =  805  cents. 


1 


f 


152  COMPOUND    NUMBEilS. 


TABLES  OF  COMPOUND  NUMBERS. 

M  0  N  E  Y. 

172.  United  States  Money  is  the  national  currency 
of  the  United  States.  The  relative  value  of  its  differ- 
ent units  or  denominations  has  already  been  given. — 
(Vide  39-43.) 

173.  English  Money  is  the  national  currency  of 
Great  Britain.  The  units  or  denominations  are  named 
Guinea,  Pound,  Crown,  Shilling,  Penny,  and  Farthing. 

TABLE. 

4  farthings  (far.)  make  1  penny,  abbreviated  d.   (denarius.) 
12  pence  "       1  shilUng,         "  s.    (solidus.) 

20  shillings  "       1  pound,  "  £.  (hbra.) 

21  shillings  "       1  guinea,  "  G. 

5  shillings  "       1  crown,  "  Cr. 

Remark  1. — The  Pound,  also  called  Sovereign,  is  the  Primary 
Unit  of  English  Money,  and  is  Avortli  $4.84.  X  Crown  is  worth 
§1.21;  a  Shilling,  $0,242;  a  Penny,  $0.0201;  a  Farthing,  G^Jj  mills. 

Remark  2. — The  Sovereign  is  22  carats  fine,  the  other  2  parts 
being  copper;  it  weighs  5  dwts.  3-J-|J-  gx*s. 

174.  French  Money  is  the  currency  of  the  Empire 
of  France.  The  Franc  is  the  unit,  and  is  worth  18 1 
cents. 

EXERCISES. 

1.  In  2  dollars  how  many  cents?  3?  4?  5?  15?  IJ? 

2.  In  200  cents  how  many  dollars?  300?  400?  500: 
1500?  150? 

3.  In  U  eagles  how  many  mills?  2?  2^?  3}? 

4.  In  15000  mills  how  many  eagles?  20000?  25000? 
32500? 


COMPOUND    NUMBERS.  153 

5.  In  2  pounds  how  many  shillings ?  3?  4?  5?  7? 
13?  15?  25? 

6.  In  40  shillings  how  many  pence?  60?  80?  100? 
140?  260?  300?  500? 

7.  In  480  pence  how  many  farthings?  720?  960? 
1200?  1680?  3120?  3600?  6000? 

8.  In  1920  farthings  how  many  pence?  2880?  3840? 
4800?  6720?  12480?  14400?  24000? 

9.  In  480  pence  how  many 'shillings?  720?  960? 
12.00?  1880?  3120?  3600?  6000? 

10.  In  40  shillings  how  many  crowns?  60?  80?  100? 
140?  260?  300'?  500? 

WEiaHT. 

175.  Troy  Weight  is  used  in  weighing  gold,  silver, 
and  precious  stones.  The  units  are  named  Grain, 
Pennyweight,  Ounce,  and  Pound. 

TABLE. 

24  grains  (gr.)  make  1  pennyweight,  abbreviated    dwt. 

20  pennyweights    "      1  ounce,  "  oz. 

12  ounces  "      1  pound,  "  lb. 

Remark  1. — The  Pound  is  the  Primary  Unit  of  Troy  Weight,  and 
is  determined  by  the  weiglxt  of  22.794422  cubic  inches  of  distilled 
water. 

JIemark  2. — The  carat  by  which  the  diamond  is  weighed  and 
valued  is  equal  to  4  grains, 

EXERCISES. 


1.  In  2  lbs.  how  many  grs.?  3? 
4?  5?  6?  13?  15?  27? 

3.  In  23  oz.  how  many  grs.? 
25?  26?  27?  28?  29? 


2.  In  11520  grs.  how  many  lbs.? 
17280?  23040?  28800?  74880? 

4.  In  11040  grs.  how  many  oz.? 
12000?  12480?  12960? 


ail 


COMPOUND    NUMBERS. 


5.|[n  1  lb.  how  many|oz.?  how 
Vmany  dwt.?  how  many  gr. ? 
7.  In  3  J  lb.  how  many  gr.  ? 


6.  In  5760  gr.  how  many  dwt.? 
how  many  oz.?  how  many  lb.? 
8.  In  20160  gr.  how  many  lb.? 


/170)  Avoirdupois  Weight  is  used  in  weighing  gro- 
ceries,  and   cheap    commodities    of  every   description. 
fThc  units   are   named  Dram,  Ounce,  Pound,  Quarter, 
Hundredweight,  and  Ton. 


abb] 

^eviatc 

)d  oz. 

a 

lb. 

u 

qr. 

ht, 

n 

cwt. 

u 

T. 

TABLE. 

16  drams"       .)  n^^ke  1  ounce, 

16  ounces.'.  -^      1  pound, 

25  pounds  ^     1  quarter, 

4  quarters  ^     1  hundredweight, 

20  hundredweight         "      1  ton. 

Remark  1. — The  Pound  is  the  Primary  Unit  of  Avoirdupois 
Weight,  and  is  determined  by  the  weight  of  27.701554  cubic  inches 
of  distilled  water. 

Remark  2. — The  Troy  Pound  is  the  same  as  \ji  lbs.  Avoirdu- 
pois, and  the  Troy  Ounce  is  the  same  as  ^||  oz.  Avoirdupois. 

EXERCISES. 


1.  In  1  T.  how  many  dr.? 
3.  In  1  cwt.  how  many  oz.? 


5.  In  17^.  how  many  dr.? 

7.  In  37  lb.  how  many  dr.? 

9.  In  47  dr.  how  many/oz.? 

11.  In  13  cwt.  how  many  qr.? 

13.  In  113  T.  how  many  dr.? 

33?  45?  127?  254? 

15.'  In  21  lb.  how  many  oz.?  23? 
17?  13?  15?  19? 


2.  In  512000  dr.  how  many  T.? 
4.  In  1600  oz.  how  many  cwt.? 
6.  In  272  dr.  how  many  oz.? 


8.  In  9472  dr.  how  many  lb.? 

10.  In  18800  oz.  how  many  qr.? 

12.  In  52  qr.  how  many  cwt.? 

14.  In  57856000  dr.  how  many  T.? 
16896000?  23040000?  G5024000? 

16.  In  336  oz.  how  many  lb.? 
368?  272?  208?  240?  304? 

^7^  At  1  cent  per  ounce,  what  v/ill  a  ton  of  raisins 
cost?  Jns.  P20.00. 

18.  At  50  cents  per  pound,  what  will  7  tons  of  butter 
cost?  Ans.  $7000.00. 


/V^ 


COMPOUND   NUxMBERS. 


155 


19.  At  5^0  cents  per  pound,  how  many  tons  of  lard 
can  be  bought  for  $35000  ?  '  ^     Ans.  85. 

177.  Apothecaries  Weight  is  used  in  mixing  medi- 
cines. The  units  arc  named  Grain,  Scruple,  Dram, 
Ounce,  and  Pound. 

TABLE. 

20  grains  (gr.)  make  1  scruple,  marked  .   .  9 

3  scruples  "      1  dram,           "  •  .  5 

8  drams  "      1  ounce,          "  •  •  ^ 

12  ounces  "      1  pound,         "  .   .  lb 

Remark. — The  Pound  is  the  Primary  Unit,  and  is  the  same  as 
the  Pound  Troy. 


EXERCISES. 


I.  In  2  ft)  how  many  §?  5?  7? 
9?  11?  13?  15? 

3.  In  2    ft)    how  many  5?    6? 
8?  10?  12?  14? 

5.  In  15  ft)  how  many  9?  17' 
19?  21?  23?  25?  29? 

7.  In  16  ft)  how  many  gr.?  18 
20?  22?  24?  2G?  28? 

9.  In  ^  lb  how  many  5?  -J?  | 

II.  In  f  of  an  §  how  many  9 

t?  F  I?  t\? 

13.  In  y'j  of  a    5    ^^^^  many 


2.  In  24  §  how  many  ft)?  60? 
84?  108?  132?  156?  180? 

4.  In  192  3  how  many  ft)?  576? 
768?  960?  1152? 

6.  In  4320  g  how  many  ft)? 
4890?  5472?  6048?  6G24?  7200? 

8.  In  23040  gr.  how  many  ft)? 
34560?  74880?  104440? 

10.  In  48  5  how  many  ft)?  32? 
24?  IG?  13f? 

12.  In  18  9  how  many  §  ?  16? 
20?  21?  10? 

14.  In  35  gr.  how  many  5?  IG? 
54?  45?  21? 


LINEAR    MEASURE. 

178.  Long  Measure  is  used  in  measuring  the  length 
of  all  quantities,  except  cloth.  The  units  are  named 
Inch,  Foot,  Yard,  Rod,  Furlong,  Mile,  and  League. 


156  COMPOUND   NUMBERS. 

TABLE. 
12     inches  (in.)  make  1  foot,  abbreviated  ft. 
"       1  yard,  "  yd. 

"      1  rod,  "  r. 

"       1  furlong,       "  fur. 

"       1  mile,  "  m. 

"      1  league,        "  1. 

IvEMARK  1. — The  Imperial  Yard  is  the  standard  of  English  linear 
measure,  and  is  determined  by  the  length  of  the  pendulum  vibra- 
ting once  a  second  at  London,  temperature  62}  degrees  Fah.  This 
length  is  39.1393  inches. 

Remark  2. — Gunter's  chain,  used  in  surveying  land,  is  4  rods 
long,  and  consists  of  100  links. 

EXERCISES. 


3 

feet 

H 

yards 

:0 

rods 

8 

furlongs 

3 

miles 

1.  In  1  m.  how  many  in.? 

3.  In  7  fur.  how  many  in.? 

5.  In  2  r.  how  many  yd.?  3? 
4?  5?  6?  75? 

7,  In  3  r.  how  many  in.?  12? 
13?  15?  17?  19? 

9.  In  1  fur.  how  many  in.? 
11.  In  1  1.  how  many  ft.? 


2.   In  63360  in.  how  many  m.? 
4.  In  55440  in.  how  many  fur.? 
6.  In    11    yd.    how    many    r.'i 
16^?  22?  27J?  33?  412}? 

8.  In   594    in.    hov/  many   r.^ 
2376?  2574?  2970?  3366?  3762? 
10.  In  7920  in.  how  many  fur.? 
12.  In  15840  ft.  how  many  1.? 


13.  How  many  inches  through  the  earth  from  pole  to 
pole?— (Vide  67,  Ex.  54.)  Ans.  500478929.28. 

14.  How  many  inches  through  the  earth  at  the  equa- 
tor? Ans.  502143776.64. 

15.  In  1  link  of  Gunter's  chain  how  many  inches? 

Ans.  7 If  inches. 

16.  In  1  mile  how  many  chains?     Ans.  80  chains. 
179.    Cloth   Measure  is  used  in  measuring  goods 

bought  or  sold  by  the  yard.  The  units  are  named  Inch, 
Nail,  Quarter,  Yard,  Ell  Flemish,  Ell  English,  and  Ell 
French. 


COMPOU^^'D   NUMBERS. 


157 


2:^  inches 
4 


ike 


nails 

quarters 

quarters 

quarters 

quarters 


TAELE. 

1     nail, 

abbrev. 

na. 

1     quarter, 

qr. 

1     yard, 

yd. 

1     ell  Flemish, 

E.  Fl. 

1     ell  English, 

E.  E.  ^ 

1  •  ell  French, 

E.  F. 

of  this    measure    is 

that    of  Long 

Remark    1. — The   stau 
Measure. 

Remark  2. — In   mercantile  practice  only   the  yard  and  quarter 
are  in  general  use. 

EXERCISES. 


1.  In  1  yd.  how  many  in.? 

3.  In  3  qr.  how  many  in.? 

5.  In  5  E.  Fl.  how  many  in.? 

7.  In  7  E.  E.  how  many  in.? 

9.  In  8  E.  Fl.  how  many  in.? 

11.  In  24  E.  Fl.  how  many  yd.? 

13.  In  70  E.  E.  how  many  yd.? 

15.  In  120  yd.  how  many  E.  FL? 

17.  In  1  E.  F.  how  many  in.? 


2.  In  30  in.  how  many  yd.? 

4.  In  27  in.  how  many  qr.? 

6.  In  135  in.  how  many  E.  FL? 

8.  In  315  in.  how  many  E.  E.? 
10.  In  216  in.  how  many  E.  FL? 
12.  In  18  yd.  how  many  E.  FL? 
14.  In  87^  yd.  how  many  E.  E.  ? 
16.  In  160  E.  FL  how  many  yd.? 
18.  In  54  in.  hoAV  many  E.  F.? 

19.  In   18360  inches  how  many   quarters?    yards? 
ells  Flemish ?  ells  English?  ells  French? 

Ans.  2040  ;  510  ;  680  ;  408  ;  340. 

20.  What  will  -|  of  a  yard  of  cloth  cost  at  7  cts.  per 
nail  ?  Atis.  98  cts. 

21.  What  will  ^^  of  a  yard  of  calico  cost  at  15  cts. 
per  nail  ?  Aiis.  12  cts. 

22.  In  2.5  feet  how  many  inches?  Aiis.  30  in. 

23.  In  3.75  furlongs  how  many  rods?       An§.  150  r. 

24.  What  is  the  number  of  miles  from  the  Equator  to 
the  North  Pole?— (Vide  67,  Ex.  61.) 

Ans.  6213.824  m. 


158 


COMPOUND    IsUMBERS. 


SUPERFICIAL    OR   SQUARE    MEASURE. 

ISO.  Square  Measure  is  used  in  measuring  sur- 
faces; as  land,  plastering,  etc.  The  units  are  named 
Square  Inch,  Square  Foot,  Square  Yard,  Square  Rod, 
Rood,  Acre,  and   Square  Mile. 


9     square  feet 
30J  square  yards 
40     square  rods 
4     roods 
G40     acres 

1  Inch. 


sq.  yd. 
sq.  r. 
R. 
A. 
M. 


TABLE. 

144     square  inches   (sq.  in.)  make  1  square  foot,  abb.  sq.  ft. 

"      1  square  yard, 
"       1  square  rod, 
"      1  rood, 
"       1.  acre, 
"       1  square  mile, 

Remark  1. — The  standard  is  the  same 
as  that  of  Long  Measure. 

Remark  2. — 16  square  rods  make  1 
i_i  square  chain,  and  10  square  chains  make 
^  1  acre. 

g  Remark  8, — The  figure  in  the  margin 
is  exactly  1  square  inch;  that  is,  it  is 
1  linear  inch  on  each  side.  144  such 
squares  are  equivalent  to  a  square  foot, 
however  the  arrangement  may  be.  They 
are  equal  to  a  square  foot  when  arranged  so  as  to  make  another 
square. 

EXERCISES. 


1  SQUARE  INCH. 


1  L\Gir. 


1.  In  1  A.  how  many  sq.  in.? 

3.  In  1  A.  how  many  sq.  r.? 
2?  3?  4?  5? 

5.  In  1  sq.  r.  how  many  sq.  ft.? 
2?  5?  8? 

7.  In  1  R.  liow  many  sq.  yd.? 
3?  13?  17? 


2.  In  6272640  sq.  in.  how  many 
A.? 

4.  In  IGO  sq.  r.  how  many 
A.?  320?  480?  640?  800? 

6.  In  272J  sq.  ft.  how  many 
sq.  r.?  544^?  1361^^?  2178? 

8.  In  1210  sq.  yd.  how  many 
R.?  3030''  ir)7:50?  20570? 


COMPOUND   NUMBER! 


159 


0.  la  1  sq.  r.  how  many  sq. 
in.? 

11.  In  1  A.  lioAv  many  sq. 
ft.? 

13.  In  1  sq.  yd.  how  many  sq. 
in.?  121?  242? 


10.  In  39204  sq.  in.  how  many 
sq.  r.? 

12.  In  130680  sq.  ft.  how  many 
A.? 

14.  In  1296  sq.  in,  how  many 
sq.  yd.?  156816?  313632? 
15.  In  252|  A.  how  many  sq.  I      16.  In  2524  sq.  ch.  how  many 
ch.?  A.? 

17.  In  -^^  m.  how  many  fur.?  18.  In  ^j  fur.  how  many  r.? 

Ans.  5^j.      I  Ans.  Z^j. 

SOLID  MEASURE. 
181.  Cubic  Measure  is  used  in  measuring  solids,  as 
timber,  earth,  and  such  other  things  as  have  length, 
breadth,  and  thickness.  The  units  are  named  Cubic 
Inch,  Cubic  Foot,  Cubic  Yard,  Ton,  Cord  Foot,  and 
Cord.        - 

TABLE. 
1728  cubic  inches  (cu.  in.)  make  1  cubic  foot,       abb.  cu.   ft. 


ton,  " 

ton  of  shipping,  " 
cord  foot,  " 

cord,  " 


cu.  yd. 

T. 

T. 

T.ofS. 

CO.  ft. 
CO. 


27  cubic  feet  "      1  cubic  yard, 

40  feet  of  round  timber      "      1  ton, 

50  feet  of  hewn  timber      "      1 

42  cubic  feet  "      1 

16  cubic  feet  "      1 

8  cord  feet  "      1 

Remark  1. — The  standard  of  this  meas- 
ure is  that  of  Long  Measure. 

Remark  2. — A  cube  is  a  solid  bounded 
by  6  equal  squares. 

Remark  3. — The  figure  in  the  margin 
represents  an  exact  cubic  inch.  Its 
squares  are  called  faces,  and  the  bounda- 
ries of  the  faces  are  called  edr/es.  Each 
edge  represents  1  linear  inch.  Each  edge 
of  a  cubic  foot  contains  12  linear  inches,  so  that  there  are 
12x12x12:    that  is,   1728  cubic  inches  in  a  cubic  foot. 


/ 

/' 

1  Cubic  Inch. 

/ 

Y   1  Linear  Inch. 

/ 

160 


COMPOU]S'D    NUMBERS. 


EXERCISES, 


1.  In  1  CO.  liow  many  cu.  ft.? 


2?  3?  4?  10? 


1-V? 


3.  In  1  CO.  liow  many  cu.  in.? 
6?  6?  7?  ^?  -I?  f? 

5.  In  3.125  CO.  liow  many  cu.  ft.  ? 
7.  In  1  cu.  yd.  how  many  cu. 


2.  In  128  cu.  ft.  how  many  co.? 
256?  384?  512?  1280?  96?  192? 

4.  In  221184  cu.  in.  how  many 
CO.?  27648?  41472? 

6.  In  400  cu.  ft.  how  many  co.? 

8.  In  46656<»cu.  in.  how  many 
cu.  yd.?  3888?  324?  36? 

182.    Wine    Measure   is   used   for   measuring   alh 
liquors,    except    ale,   beer,    and   milk.     The   units    are 
named  Gill,  Pint,  Quart,  Gallon,  Tierce,  Barrel,  Hogs- 
head, Pipe,  and  Tun. 

TABLE. 
4     gills  (gi.)  make  1  pint,  abbreviated  pt. 


2  pints 

4  quarts 

31J-  gallons 

42  gallons 

63 

2 


gallons 
hogsheads 


pipe 


1  quart,  " 

1  gallon,  " 

1  barrel,  " 

1  tierce,  ^" 

1  hogshead,  " 

1  pipe,  " 

1  tun.  ^ 


qt. 

gal. 

bbl. 

ti. 

hhd. 

py 


Remark. — The  Wine- Gallon  contains  231  cubic  inched 


1.  In  1  tun  how  many  hhd.? 
gal.?  qt.?  pt.?  gi.? 

3,  In  13  gal.  how  many  gi.? 
15?  17?  19?  21?  23? 

6.  In  5  tuns  how  many  gi.?  7? 
9?  11?  13?  17? 

7.  In  3  bbl.  how  many  gal.? 
4?  7?  10?  13?  16? 

9.  In  1  hhd.  how  many  bbl.? 
11.  In  126  ti.  how  many  hlid.? 
13.  In  2.5  qt.  how  many  gal.? 
15.  In  23.625  gal.  how  many  hhd  ? 


EXERCISES. 

2.  In  8064  ^\A\( 
pi 


(ii€ ,  -1  i 


hhd.?  pi.?   tuns? 

4.  In  416  gi.  how  many  gal.? 
480?  544?  608?  072?  ^? 

6.  In  40320  gi.  how  m^y=  tuns/ 
56448?  72-576?  88704? 

8.  In  94J.,gal.  how  many  bbl.? 
126?  220i'j^l5?  409 J? 
10.  In  1«"0  bbl.  how  many  hhd.? 
12.  In  168  bbl.  how  many  ti.? 
14.  In  .625  gal.  how  many  qt.? 
16.  In  .375  hhd.  how  many  gnl.? 


3^ 


COMrOUXD    XUxMBERS.  161 

183.  Ale  or  Beer  Measure  is  used  for  measuring 
ale,  beer  and  milk.  The  units  are  named  Pint,  Quart, 
Gallon,  Barrel,  and  Hogshead. 

table. 

2  pints  (pt.)  make  1  quart,  abbreviated  qt. 

4  quarts  "  1  gallon,  "       '    gal. 

36  gallons  "  1  barrel,  "  bbl. 

\\  barrels         .    "  1  hogshea'd,      "  hhd. 

Remark. — The  Beer  Gallon  contains  282  cubic  inches. 
EXERCISES. 


1.  In  I  hhd.  how  many  gal.? 
qt..?  pt.?  / 

3.  In  693  Beer  Gal.  how  many- 
Wine  Gal.?  2079? 

6.  In  Jg  of  a  gal.  how  many  pt.  ? 

7.  In  -^-^  of  a  hhd.  how  mamy 
pt.?        '  '"^ 

9.  In  \  of  a  bbl.  how  many 
qt.? 


2.  In  432  pt.  how  many  qt.? 
gal.?  bbl.? 

4.  In  846  Wine  Gal.  how  many 
Beer  Gal.?  2538? 

6.  In  I  a  pt.  how  many  gal.? 

8.  In  1  pt.  what  part  of  a 
hhd.? 

10.  In  36  qt.  what  part  of  a 
bbl.? 


184.  Dry  Measure  is  used  in  m^easuring  such  arti- 
cles as  grain,  fruit,  etc.  The  units  are  named  Pint, 
Quart,  Peck,  Bushel,  and  Quarter. 

TABLE. 

2  pints  (pt.)  make  1  quart,  abbreviated  qt. 

8  quarts  "       1  peek,             "          pk. 

4  pecks  "       1  bushel,          "          bu. 

8  bushels  "       1  quarter,         "          qr. 

Remark. — The  Winchester  Bushel  is  a  cylinder,  18^-  inches 
internal  diameter,  and  8  inches  deep.  It  contains  2150.4  cubic 
inches. 

14 


162  COMPOUJS^D   XUMBEilS. 

EXERCISES. 

1.  What  cost  25  quarters  of  wheat  at  90  cents  per 
bushel?  A71S.  $180. 

2.  At   90    cents    per  bushel  how  many  quarters  of 
wheat  can  be  bought  for  $360  ?         Ans.  50  quarters. 

3.  What  cost  17  bushels  of  apples  at  27  cts.  a  peck? 

4.  At  27  cents  a  peck,  how  many  bushels  of  apples 
can  be  bought  for  $18.36  ? 

5.  What  must  be  paid  for  7  bushels  of  chestnuts  at 
3  cents  a  pint?  Ans.  $13.44. 

6.  What  cost  I  of  a  pint  of  blackberries  at  $3.20  per 
bushel?  Ans.  3  cents. 

7.  W^hat  cost  25-J  bushels  of  potatoes  at  20  cents  a 
peck?  ^ns.  $20.70. 

8.  How    many   bushels    of  potatoes   can    I  buy  for 
$41.40,  at  2-J  cents  a  quart?  Ans.  51|-  bushels. 

TIME. 

185.    The  units  of  Time  are  named  Second,  Minute, 
Hour,  Day,  Week,  Month,  Year,  Century. 

TABLE. 

GO  seconds  (sec.)  make  1  minute,  abbreviated  m. 

60  minutes  "      1  hour,  "  h. 

24  hours  "      1  civil  day,     "  d. 

%  days  "      1  week,  "  w. 

12  months  "      1  year,  "  y. 

Remark  1. — The  standard  unit  of  time  is  the  period  occupied  by 

the  eurth  in  making  one  revolution  on  its  axis,  wliich  period  is 

called  a  Sidereal  Day  and  consists  of  23  h.  5G  m,  4  sec. 

Remark  2.— The  Tropical  Year  consistR  of  305  d.  5  h.  4S  m.  47.57 

S3C. 


COM IHJ  V .\ D    N  U  M 13 ERS . 


163 


REMAini:  3.— The  Civil,  Legal,  or  Julian  Year  consists  of  365 
days,  except  Leap  Year,  "tvhich  consists  of  300  days. 

Remark  4. — Every  year  which  is  exactly  divisible  by  4  is  a 
Leap  Year,  excepting  those  centennial  years  not  exactly  divisible 
by  400.  Thus,  1868  will  be  a  Leap  Year;  1900  will  not  be  a  Leap 
Year;  but  the  year  2000  will  be  a  Leap  Year. 


TABLE    OF    THE   MGNTIH 


Mouth. 

Abb. 

Order. 

Xo.  D. 

Mouth. 

Abb. 

Order. 

Xo.  D. 

January. 

Jan. 

1st. 

31. 

July. 

7th. 

31. 

February. 

Feb. 

2d. 

28. 

August. 

Aug. 

8th. 

31. 

March. 

Mar. 

3d. 

31. 

September, 

Sept. 

9th. 

30. 

April. 

Apr. 

4th. 

30. 

October. 

Oct. 

10th. 

31. 

May. 

5th. 

31. 

November. 

Nov. 

11th. 

30. 

June. 

6th. 

30. 

December. 

Dec. 

r2th. 

31. 

Remark  5. — In  Leap  Year,  February  has  29  days. 
Remark  6. — In  finding  the  interval  between  two  dates,  it  is 
customary  to  consider  the  months  as  having  30  days  each. 

TABLE 

Showing  the  time  in  days  from  any  day  in  one  month  to  the  corref^pond- 
ing  day  in  another  month. 


January... 
February... 

March 

April 

Mav 


June 

•'^ny 

AugXTSt 

Sepr'ember. 

October 

November . 
December.. 


365 
1 334 

■306! 

!275| 

i245 

l214! 

184 

153 

122 

92 

61 

31 


31 
365 
337 
306 


59 

28 

365 

334  365 
276|304l335 
245I273!304 
21o;2!3274 
184  212  243 


89 

61 

30 

365 


90;120il51181 

59 

31 


120  150 


1334  365 
304  335 


122 
91 
61 
30 

365 


212243 
1811212 
153;184 
122  153 


t\^ 


153 

123 

92 

62 


181J212 

151  i  182 

120;i51 

90:i21 


242!273!303 
2121243:273 
18l!212;242 
151:182212 


92 

61 

31 

365 

334 


123 
92 
62 
31 

365 


2731304  334 
242i273  303 
214|245j275 
183l214|244 
!153  1841214 


153{1 


122 

92 

61 

30 
[365 
13341365 


3041335 
273|304 
243:274:3041 3351 365 


1231153 

92J122 

61    91 

3l|  61 

30 


164  COMPOUND   NUMBERS. 

EXERCISES. 

1.  How  many  days  from  January  10th  to  June 
10th?  Ans,  151. 

Find  January  in  the  left-hand  column,  and  follow  the  line  to 
the  right  till  you  come  to  June, 

2.  How  many  days  from  February  6th  to  May 
6th?  Ans.  89. 

3.  How  many  days  from  January  1st  to  July 
4th?                                   ■  Ans.  184. 

Here  add  3  to  the  tabular  number,  which  is  181. 

4.  How  many  days  from  December  25th  to  July 
10th?  Ans.  197. 

Here  subtract  15  from  the  tabular  number,  212, 

5.  How  many  days  from  January  17,  1868,  to  July 
17,  1868?  Ans.  182. 

Here  add  1  to  the  tabular  number  for  Leap  Year,  as  the  dates 
include  the  month  of  February, 

CIRCULAR    MEASURE. 

186.  Circular  Measure  is  used  in  estimating  Lati- 
tude and  Longitude,  and  in  measuring  the  relative  dis- 
tances of  the  Planets  and  other  heavenly  bodies.  The 
units  are  named  Second,  Minute,  Degree,  Sign,  and 
Circumference  of  Circle. 

TABLE. 

60  seconds  (")  make  1  minute,  marked  ' 
60  minutes  "      1  degree,         "        ° 

30  degrees  "      1  bign,  abbreviated  S. 

12  signs  make  1  circumference  of  circlcj  abb.  circ. 


COMPOUND    NU3IBE11S. 


166 


Remark  1. — The  length  of  a  degree  measured  on  the  equatoi-  is 
69.161  m.  The  length  of  a  degree  measured  on  a  meridian  is 
69.042  m. 

Remark  2. — A  degree  contains  60  geo- 
graphic miles. 

Remark  3. — Since  every  circumference  of 
a  circle  contains  360  degrees,  the  length  of 
the  degree  varies  as  the  diameter  of  the  cir- 
cle varies.  The  circumference  of  a  circle 
is  always  about  3.1416  times  its  diameter. 

Remark  4. — The  Celestial  Equator  is  di- 
vided into  12  Signs;  their  names  are  Aries, 
Taurus,  Gemini,  Cancer,  Leo,  Virgo,  Libra,  Scorpio,  Saggitarius, 
Capricornus,  Aquarius,  and  Pisces. 


EXERCISES. 

1.  In  1  circle  how  many  degrees?  minutes?  seconds? 

2.  What  is  the  length  of  1'  along  the  equator? 

AiiB.  1.15268  m. 

3.  What  is  the  length  of  V  along  a  meridian  ? 

Am.  1.1507  m. 

4.  What  is  the  length  of  V  along  the  equator? 

An%.  .01921. 

5.  What  is  the  length  of  V^  along  a  meridian  ? 

Am,  .01918  nearly. 

6.  If  the  semi-circumference  of  a  circle  is  3.1415926- 
535897932,  etc.,  inches  in  length,  as  it  really  is  when 
the  diameter  is  2  inches,  what  is  the  length  of  1'  ? 

Am.  .000290888298665721  in. 

7.  What  is  the  circumference  of  a  circle  whose  diam- 
eter is  10  inches?  Am.  31.4159  in. 

8.  What  is  the  circumference  of  a  circle  whose  diam- 
eter is  10,000,000  miles? 

Am.  31415926.535  m. 


166 


COMPOUND   ^'UMBERS. 


KEDUCTION  OF  COMPOUND  NUMBERS. 

187.  Reduction  in  Aritlimetic  consists  in  making 
some  change  in  the  method  of  i^epresenting  a  quantity. 
Hence,  reduction  makes  no  change  upon  the  value  of  a 
quantity. 

188.  It  has  been  seen  that  many  simple  concrete 
numbers  may  be  reduced  to  other  concrete  numbers  of 
a  lower  unit  value  By  multijMcation,     Thus, 

2 £=40 s., because  20x2=40.  (Vide  174,  Ex.  5.) 

189.  *lt  has  also  been  seen  that  simple  concrete 
numbers  may  be  reduced  to  other  simple  concrete  num- 
bers of  a  higher  unit  value  by  division.     Thus, 

72  in.=  6  ft.,  because  72-f-12=6.  (Vide  178.) 


190-  To  reduce  a  compound 
number  to  a  simple  concrete 
number, 

Arrange  the  different  units  com- 
posing the  compound  number  in  a 
horizontal  line^  and  over  each  place 
the  number  connecting  it  with  the 
next  higher  unit. 

Multiply  the  units  of  the  highest 
value  hy  that  number  which  stands 
over  the  next  lower  units,  and  to  the 
])roduct  add  the  same  loiver  units; 
then  multiply  the  sum  by  the  num- 
ber standing  over  the  next  lower 
units,  etc.,  continuing  the  work  till 
the  lowest  units  given  have  been 
added. 

EXAMPLES. 

1.  Reduce  17 £  68.  Od.  8  far.  to 
fartliinss. 


191.  To  reduce  a  simple  con- 
crete number  to  a  compound 
number. 

Divide  the  given  number  by  that 
number  which  connects  it  ivith  the 
unit  of  the  next  higher  value,  plac- 
ing the  remainder,  if  there  be  any, 
to  the  right. 

Continue  the  work  till  the  unit 
of  the  highest  value  is  reached,  or, 
till  the  next  divisor  would  be  greater 
than  the  dividend. 

Ulc  last  quotient  and  the  several 
remainders,  written  in  the  order  of 
their  tmit  values,  will  be  the  com' 
2^ound  number. 

E  X  A  ai  P  L  E  S  . 

2.  Reduce  1G599  farthings  to 
pounds,  etc. 


CO M PO  L  X D    >:  U:.I BERS . 


167 


b. 

7. 

9. 
11. 
grai 
13. 
15. 
17. 
19. 
21. 
23. 
25. 


OPERATION. 
20     12      4 

17     5     9     3 
20 

345 

12 

4149 
4 

16599  far.  Ans. 

Reduce  20  <£  15  s.  9d.  3  far. 
Reduce  240£  Os.  7d.  2 far. 
Reduce  17s.  Ifar.  to  far. 
Reduce  lib.  loz.ldwt,  Igr. 
Reduce  17  lb.  5  dwt.  to 
ns. 

Reduce  Soz,  3  dwt.  3gr. 
Reduce  7  T.  15  oz.  to  oz. 
Reduce  3T.  7cwt.  3qr. 
Reduce  3qr.  11  oz.  13  dr. 
Reduce  1  lb  3  §  55. 
Reduce  lib  2  3  to  grains. 
Reduce  5r.  4  yd.  2  ft.  7  iu. 

OPERATION. 
5K    3     12 


5    4    2     7 
54 


3U 
3^ 


96^ 
12 


OPERATION. 


1G599 


12 


20 


4149—3 


345—9 


4. 

6. 

8. 
10. 
12. 
etc. 
14. 
16. 
18. 
20. 
22. 
24. 
26. 


17—5 
17-£  5s.  9d.  3far.  Ai 


Reduce  19959  far.  to  £,  etc. 
Reduce  230430  far.  to  £. 
Reduce  817  far.  to  s.,  etc. 
Reduce  6265  gr.  to  pounds. 
Reduce  98040 gr.  to  pounds, 

Reduce  1515gr.  to  oz.,  etc. 
Reduce  224015  oz.  to  T.,  etc. 
Reduce  27;igr.  to  T.,  etc. 
Reduce  19389  dr.  to  qr.,  etc. 
Reduce  7500 gr.  to  To,  etc. 
Reduce  5800 gr.  to  lb,  etc. 
Reduce  1165 in.  to  r.,  etc. 

OPERATION. 


1165 
97—1 
32—1 


64  =lialf  yds. 
5— 4  J 

1165  iry.AriS.  ^/Js.  5r.  4)/^yd.lft.lin.=5r.  4j'd.2ft.7m. 

27.  Reduce  401.  6  fur.  2  in.  to  I      28.    Reduce     7650722    in.    to 
inches.  |  leagues,  etc. 

29.  Reduce    22  fur.    IGr.  3yd.  |      30.  Reduce    1479-4 ft.    to    fur- 

1ft.  to  ft-et.  longs,  etc. 


168 


31.  Reduce  4  m.  7  fur.  20 r  16 
ft,  to  inches. 

33.  Reducel  yd.  Iqr,  2na.  7}in. 

35.  Reduce  4  tuns  5hhd.  3qt. 
to  quarts. 

37.  Reduce  Itun  Igal.  3qt.  to 
gills. 

39.  Reduce  47bbl.  18  gal.  of 
ale  to  pints. 

41.  Reduce  15  bu.  2pk.  7qt.  to 
quarts. 

43.  Reduce  9bu.  5qt.  Ipt.  to 
pints. 

45.  Reduce  14  A.  IR.  17  r.  to 
rods. 

47.  Reduce  17  A.  3R.  12  r.  to 
square  feet. 

49.  Reduce  3  da.  55  m.  to  min- 
utes. 

51.  Reduce  9S.  13°  25^  to  sec- 
onds. 

63.  Reduce  25°  14^  V^  to  sec- 
onds. 

55.  Reduce  5  fur.  3  r.  10  ft.  6  in. 
to  inches. 

57.  Reduce  21$  3  m.  to  mills. 


COMPOUND   NUMBERS. 

32.  Reduce  313032  in.  to  miles, 


etc. 


34.  Reduce  50 in.  to  yards,  etc. 
36.  Reduce    5295  qt.    to    tuns, 


etc. 

38.  Reduce  8120  gi.  to  tuns, 
etc. 

40.  Reduce  13680pt.  to  barrels, 
etc. 

42.  Reduce  503  qt.  to  bushels, 
etc. 

44.  Reduce  587  pt.  to  bushels, 
etc. 

46.  Reduce  2297  r.  to  acres, 
etc. 

48.  Reduce  776457  sq.  ft.  to 
acres,  etc. 

50.  Reduce  4375m.  to  days, 
etc. 

52.  Reduce  1020300^^  to  signs, 
etc. 

54.  Reduce  90847^''  to  degrees, 
etc. 

56.  Reduce  40320 in.  to  fur- 
longs. 

68.  Reduce  21003  m.  to  dollars. 


59.  What  cost  1  lb.  1  oz.  1  dwt.  1  gr.  of  gold,  at  3^ 
cts.  per  grain?  Ans.  208.83 J. 

60.  If  gold   is  worth  3 J  cts.  per. grain,  hoy>^  many 
pounds  can  be  bought  for  $626.50  ? 

Ans.  3  lb.  3  oz.  3  dwt.  3  gr. 

61.  What   cost  17  lb.   5  dwt.  of  silver,  at  31^  cts. 
per  dwt.  ? 

62.  What  weight  of  silver  can  be  bought  for  |1260.- 
93|-,  at  the  rate  of  31 J  cts.  per  dwt? 


COMPOUXD    NUMBERS.  1G9 

63.  What  will  3  T.  7  cwt.  3  qr.  of  rice  cost  at  3d. 
English  money  per-  pound  ? 

64.  How  much  rice  at  3d.  per  pound  can  be  bought 
for  £84  13s.  9d.? 

65.  At  the  rate  of  |0.060J  per  pound,  how  many  tons 
of  hay  can  be  had  for  $4D9.887i  ? 

66.  What  cost  4  m.  7  fur.  20  r.  16  ft.  of  railroad,  at 
an  expense  of  |3.78f|  per  ft.? 

67.  How  many  miles  of  railroad  can  be  built  for 
^98810.60f  §,  at  an  expense  of  3f  |  dollars  per  ft.  ? 

68.  What  will  4  tuns  5  hhd.  3  qt.  of  molasses  cost, 
at  15  cts.  per  qt.  ? 

69.  What  number  of  tuns  of  claret  can  be  had  for 
$794.25,  at  the  rate  of  7J  cts.  per  pt,? 

70.  What  will  15  bu.  2  pk.  7  qt.  of  chestnuts  amount 
to  in  dollars,  at  3  farthings  English  money  per  pt.  ? 

71.  How  many  bushels  of  chestnuts  can  be  had  for 
§15.215-1  at  the  rate  of  IJd.  per  qt.? 

72.  If  a  rod  of  land  produce  3i  pounds  of  cotton, 
how  many  bales  can  be  taken  from  14  A.  1  R.  17  r.  of 
land,  at  500  pounds  to  the  bale  ? 

73.  If  a  rod  of  land  produces  3|  pounds  of  cotton, 
what  number  of  acres  will  produce  14yYo  bales  ? 

74.  A  gentleman  having  17  A.  3  R.  10  r.  of  land,  laid 
it  off  in  lots,  each  containing  25  rods,  and  sold  them  at 
|450  per  lot.     How  much  did  he  get  for  his  land? 

75.  If  I  sell  a  quantity  of  land  at  the  rate  of  $450 
for  25  sq.  r.,  and  obtain  $51300,  how  many  acres  do  I 
sell? 

192.  To  reduce  a  denominate       103.  To  rediioe  a  compound 

fraction  to  a  compound  number,    number  to  a  denominate  fraction, 

Multiply  the  fraction  by  thenum-       Reduce  the  unit  of  the  proposed 

15 


170 


COMPOUND    NUMBERS. 


her  zvhich  connects  it  with  the  next 
loicer  imitj  and  the  fractional  part 
of  the  product  by  the  number  which 
connects  it  with  the  next  lower  unit, 
and  so  on  till  the  lowest  unit  of  the 
table  is  reached. 

The  several  integral  parts  of  the 
product  will  form  the  compound 
number,  retaining  the  fractional 
part,  if  there  be  any,  of  the  last 
quotient. 


EXAMPLES. 


compound  number. 


OPERATION. 


11)56(5         (Vide  180,  Ex.  17.) 
65 

1 

40 

11)40(3         (Vide  180,  Ex.  18.) 


7 

m 

ll)115i(10 

no 


5J 
12 


11)66(6 
66 


Ans.  5 fur.  3r.  lOft.  Gin. 


fraction  to  the  lowest  units  men- 
tioned in  the  compound  number  for 
the  denominator  of  the  required 
fraction. 

Reduce  the  compound  number  to 
the  same  units  for  the  numerator  of 
ths  fraction. 

Reduce  the  fraction  to  its  lowest 
terms. 

Remark. — If  there  is  a  fraction 
connected  loith  the  lowest  units,  mul- 
tiply both  parts  by  the  denominator 
of  the  fraction  before  reducing. 


examples. 

2.  Reduce  5  fur.  3r.  10  ft.  6  in. 

to  a  denominate  fraction. 

operation. 

40 

16>^  12 

1 

8 

5     3 
40 

10    6 

8 
40 

203 
16J 

320 
16^ 

3359* 
12 

5280 
12 

40320 

63360 

(Vide  178,  Ex.  1,  and  190,  Ex.55.) 


COMPOUND    NrMBERS. 


171 


3.  Reduce  |ra.  to  a  compound 
number. 

^   5.  Reduce    y'gCwt.    to    a    com- 
pound number. 

7.  Reduce  f  T.  to  a  compound 
number. 

9.  Reduce  ^£  to  a  compound 
number. 


11.  Reduce  |s.  to  a  compound 
number. 

13.  Reduce  ^d.  to  a  compound 
number. 

15.  Reduce  iituns  to  a  com- 
pound number. 

17.  Reduce  flilid.  to  a  com- 
pound number. 

19.  Reduce  f  bbl.  wine  to  a 
compound  number. 

21.  Reduce  If  A.  to  a  com- 
pound number. 

23.  Reduce  17|fA.  to  a  com- 
pound number. 


4.  Reduce  3 fur.  22 r.  3  ft.  Sin. 
to  fraction  of  mile. 

6.  Reduce  Iqr.  181b.  12  oz.  to 
fraction  of  hundredweight. 

8.  Reduce  8cwt.  2qr.  71b.  2  oz. 
4idr.,  etc. 

10.  Reduce  8s.  6  d.  3ffar.,  etc. 


6     3f 


OPERATION. 

OPEEATION. 

3 

20 

1 
20 

8 

12 

7)60(8 
66 

20 
12 

102 
4 

4 

240 

411f 

12 

4 

2880 

7)48(6 
42 

960 

7 

6 

6720 

(191  Rem.) 

4 

|f|^=3£  Ans. 

7)74(3^ 

' 

8s.  6d.  33  far.  Ans. 

12.  Reduce  4d.  2  far.  to  frac- 
tion of  a  shilling. 

14.  Reduce  If- far.  to  fraction 
of  a  penny. 

16.  Reduce  Ipi.  Ihhd.  42  gal. 
to  fraction  of  a  tun. 

18.  Reduce  23gal.  2qt.  Ipt., 
etc. 

20.  Reduce  18 gal.  3qt.  Ipt. 
f  gi.,  etc. 

22.  Reduce  1  A.  IR.  28fr.  to 
fraction  of  acre. 

24.  Reduce  17  A.  3R.  12  r.  to 
fraction  of  acre. 


172 


COMPOUND  XL'MEERS. 


25.  Reduce  ^-^  yr.  to  a  com- 
pound number. 

27.  Reduce  |-  w.  to  a  compound 
number. 

29.  Reduce  |  h.  to  a  compound 
number. 

31.  Reduce  ^bu.  to  a  compound 
number. 

33.  Reduce  ||  bu.  to  a  com- 
pound number. 

35.  Reduce  3|  lb.  Troy  to  a 
compound  number. 


2(i.  Reduce    TOO 
fraction  of  year. 


da.    12  h.    to 


28.  Reduce  2  da.  19  h.  12  m.  to 
fraction  of  week. 

30.  Reduce  22  m.  30  sec.  to 
fraction  of  hour. 

32.  Reduce  3  pk.  2qt.  IJpt.  to 
fraction  of  bushel. 

34.  Reduce  1  pk.  6  qt.  §  pt.  to 
fraction  of  bushel. 

36.  Reduce  3  lb.  10  oz.  to  frac- 
tion of  pound, 

38.  Reduce  9  dwt.  9  gr.  to  frac- 
pound  number.  tion  of  pennyweight. 

39.  If  I  ride  5  fur.  3  r.  10  ft.  6  in.  in  a  railroad  car, 
what  ought  to  be  my  exact  fare  at  the  rate  of  11  cts.  a 
mile  ?  Ans.  7  cts. 

40.  What  cost  the  iron  on  a  track  measuring  3  fur. 
22  r.  3  ft.  8  in.,  at  the  rate  of  §4500  per  mile  ? 

Ans.  §2000. 

41.  Sold  1  pi.  1  hhd.  42  gal.  of  molasses  at  the  rate 
of  175.60  per  tun.     What  did  I  get  ?       Ans.  $69.30. 

42.  A  grocer  bought  8  cwt.  2  qr.  7  lb.  2  oz.  44  dr.  of 
coffee  at  §9.50  per  cwt.,  and  sold  it  at  a  retail  price  of  16 
cts.  per  lb.     How  much  did  he  make?    A7is.  §55.71  f. 

43.  Bought  1  pk.  6  qt.  f  pt.  of  chestnuts  at  the  rate 
of  §2.88  per  bushel,  and  retailed  them  at  9  cts.  per 
quart.     Do  I  make  or  lose  ? 

44.  If  a  boy  could  count  6000  marbles  in  an  hour, 
how  many  could  he  count  in  22  m.  30  sec.  ? 

Ans.  2250. 

45.  If  I  buy  iron  at  §45  per  ton,  and  sell  it  at  2J  cts. 
per  pound,  what  do  I  gain  by  selling  13  cwt.  2  qr.  15 
lb.?  Ans.^S.41h 


COMPOUND   NUMBERS. 


173 


194-  To  reduce  a  compound 
number  to  a  decimal  fraction, 

Divide  the  lowest  units  by  that 
number  which  connects  them  with 
the  next  higher^  and  annex  the  quo- 
tient as  a  decimal  to  the  given  num- 
ber of  those  higher  units. 

Continue  to  divide  till  the  units 
required  are  reached. 

The  last  quotient  will  be  the  re- 
quired decimal. 

EXAMPLES. 

1.  Reduce  23  gal.  2  qt.  1  pt.  to 
a  decimal  in  hogshead. 


OPERATION. 


2.5.      (Vide  182,  Ex.  13. 


63123.625    (Vide  182,  Ex.  15.) 
.375  hhd.  Ans. 
(Vide  192,  Ex.  17.) 


3,  Reduce  3  fur.  22  r.  3  ft.  8  in. 
to  decimal  fraction  of  mile. 

5.  Reduce  1  qr.  18  lb.  12  oz.  to 
fraction  of  hundredweight. 

7.  Reduce  8  s.  6d.  3^  far.  to 
fraction  of  pound  sterling, 

9.  Reduce  15  £  10s.  9d.  to 
fraction  of  pound  sterling. 

11.  Reduce  18  h.  9  m.  to  frac- 
tion of  day. 

13.  Reduce  5  cwt.  2  qr,  15  lb.  to 
fraction  of  ton. 


195.  To  reduce  a  denominate 
decimal  fraction  to  a  compound 
number. 

Multiply  the  given  decimal  frac- 
tion by  the  number  which  connects 
it  with  the  next  lower  units,  and  the 
decimal  part  of  the  product  by  the 
number  connecting  it  tvith  the  next 
lower  units,  and  so  on  till  the  lowest 
unit  is  reached.  The  integral  parts 
of  the  products  will  form  the  num- 
ber required. 

EXAMPLES. 

2.  Reduce  .375  hhd.  to  a  com- 
pound number. 

OPERATION. 

.375  hhd. 
63 


23.625  gal. 
4 


2,500  qt. 
2 


1.000     pt. 
Ans.  23  gal.  2qt.  Ipt. 

4.  Reduce   .44|  m   to   a   com- 
pound number. 

6.  Reduce  .4375  cwt.  to  a  com- 
pound number. 

8.  Reduce  .42857^ £  to  a  com- 
pound number. 

10.  Reduce  15.5375 <£  to  a  com- 
pound number. 

12,  Reduce  .75625  da.  to  a  com- 
pound number. 

14.  Reduce  .2825  T.  to  a  com- 
pound uumbci'. 


174:  COMPOUND    N^UMBERS. 


15.  Reduce  3  ft.  9  in.  to  frac- 
tion of  yard. 

17.  Reduce  3  pk.  2  qt.  lipt.  to 
fraction  of  bushel. 

19.  Reduce  22  m.  30  sec.  to 
fraction  of  hour. 

21.  Reduce  2  da.  19  h.  12  m.  to 
fraction  of  week. 

23.  Reduce  3°  30^  36^^  to  frac- 
tion of  degree. 


16.  Reduce  1,25  yd.  to  a  com- 
pound number. 

18.  Reduce  .83^  bu.  to  a  com- 
pound number. 

20.  Reduce  .375  h.  to  a  com- 
pound number. 

22.  Reduce  .4  w.  to  a  compound 
number. 

24.  Reduce  3.51°  to  a  compound 
number. 


25.  What  will  23  gal.  2  qt.  1  pt.  of  wine  cost  at  $60 
per  hhd.  ?  A71S.  $22.50. 

26.  What  will  the  above  wine  bring  at  $1  per  gal.  ? 

Alls.  §23.625. 

27.  What  will  3  pk.  2  qt.  IJ  pt.  of  corn  cost  at  90 
cts.  per  bushel  ?  Ans.  75  cts. 

28.  If  I  buy  13  A.  2  R.  35  r.  of  land  at  $17.28  per 
acre,  and  sell  it  in  lots  of  1  rood  each  at  12  dollars  a 
lot,  how  much  do  I  make?  Ans.  $421.44. 

29.  What  will  .2825  tons  of  rice  bring  at  7  cts.  per 
lb.?  at  $1.50  per  qr.?  at  $5  per  cwt.  ? 

Ans.$S9.bD;  $33.90;  $28.25. 

30.  At  $25  per  acre,  how  much  land  can  be  bought 
for  $648.75  ?  Aits.  25.95  A.-=25  A.  3  R.  32  r. 

31.  Bought  18  cwt.  1  qr.  18  !b.  of  tea  at  $65  per  cwt., 
and  sell  the  same  at  75  cts.  per  lb.  How  much  do  I 
I  make?  Ans.  $184.30. 

32.  Bought  56  hhd.  16  gal.  3  qt.  of  molasses  at 
$46  per  hhd.     What  did  it  amount  to? 

Alls.  $2588.23. 

33.  I  sell  the  above  molasses  at  87J  cts.  per  gal. 
What  do  I  gain  ?  Ans.  $513.43. 

34.  A  planter  sold  270  bales  of  cotton  at  $60  per 


COMPOUND    NUMBERS.  175 

bale,  and  invested  the  proceeds  in  land  at  $28  per  acre.  , 
How  much  land  did  he  purchase  ? 

Ans.  578  A.  2  R.  llf  r. 

35.  A  merchant  imported  325  yards  of  silk,  at  an 
expense  of  £1  4s.  6d.  per  yard,  and  desires  to  clear 
$1200  in  retailing  it.  What  must  be  the  price  per  yard  ? 
(Vide  173,  Rem.  1.)  A7is.  $9,621. 

36.  I  import  275  yards  of  French  broadcloth,  at  an 
expense  of  24  francs  per  yard,  and  clear,  in  retailing  it, 
$500.     What  do  I  charge  per  yard?     (Vide  174.) 

Ans,  $6.23. 

ADDITION   OF    COMPOUND    NUMBERS. 

190.    To  add  compound  numbers, 

(1.)  Wrile  the  units  of  the  same  name  under  each  other ^ 
and  place  over  each  column  that  number  which  connects 
its  unit  with  the  next  higher  unit. 

(2.)  Add  the  coluynn  of  units  of  the  least  value,  and 
divide  the  sum  hy  that  number  which  stands  over  it, 
placing  the  remainder  under  the  column. 

(3.)  Add  the  column  of  units  of  the  next  higher  value, 
including  the  quotient  of .  the  preceding  division,  and 
divide  by  the  number  which  stands  over  it,  placing  the  re- 
mainder under  the  column. 

(4.)  Add  all  the  colmnns  of  units  in  the  same  ivay, 
writing  down,  hoivever,  the  entire  sum  of  the  column 
ivhich  can  have  no  number  over  it. 

'     EXAMPLES. 

1.  Add  together  20£  15s.  9d.  3  far. ;  240c£  7d.  2  far.; 
and  17s.  1  far. 


176  COMPOUND    NUMBERS. 

2.  Add  together  40  1.  6  fur.  2  in.;  2  m.  6  fur.  16  r. 
3  yd.  1ft.;  4  m.  7  fur.  20 r.  16  ft.;  11.  2  m.  7  fur.  13  r. 
1  ft.  11  in. 


OPERATIONS. 

(1-)  (2.) 


20 

12 

4 

20 

15 

9 

3 

240 

0 

7 

2 

17 

0 

1 

40      5%  3        12 


40     0 

6 

0 

0 

0 

2 

2 

6 

16 

3 

1 

4 

7 

20 

0 

16 

1      2 

7 

13 

0 

1 

11 

Ans.  261  £  13  s.  5d.  2  far. 

441.  2  m.  3  fur.  10  r.  3  ^yd.  1ft.  1  in. 

Ans.  441.  2  m.  3  fur.  10  r.  3  yd.  2  ft.  7  in.,  since  ^  yd.=l  ft.  6  in. 

3.  Add  together  1  hhd.  25  gal.  3  qt.  1  pt. ;  8  hhd.  2  qt. ; 
3  hhd.  27  gal.  1  pt. ;  and  21  hhd.  and  1  pt. 

Ans.  8  T.  1  hhd.  53  gal.  2  qt.  1  pt. 

4.  Add  together  13  bu.  2  pk.  7  qt.  1  pt. ;  150  bu.  1  pk. 
5  qt.;  200  bu.  3  pk.  5  qt.  1  pt.  Ans.  365  bu.  2  qt. 

5.  Add  together  J  J  of  a  tun,  §  of  a  hhd.,  and  |  of  a 
bbl. 

OPERATION. 

2         2  f)3  4  2 


\i  tun  = 
1  hhd.- 

f   bbl.=: 

1     1     42 
23 

18 

0 
2 
3 

0  (Vide  192,  Ex.  15.) 

1  (    "    192,   "    17.) 
li    (    "    192,   "    19.) 

ItunO     0     21  gal.  2  qt.    J  pt. 

6.  Add  together  |  £  g  s.  and  f  d. 

Ans.  8s.  lid.  3|far. 

7.  Add  together  1|  A.  and  17f  §  A. 

Ans.^lO  A.  1  R.  4  r. 

8.  Add  together  /jj  yr.  §  w.  and  |  h. 

Ans.  112  da.  7  h.  34  m.  30  sec. 


COMPOUND    NUMBERS.  177 

9.  Add  together  §  bu.  and  ||  bu. 

Ans.  1  bu.  1  pk.  1  qt. 

10.  Add  together  j  i  m.  and  f  fur. 

Ans.  7  fur.  26  r.  5  ft.  4,%  in. 

11.  Add  together  15.5375£  and  .4285|£.  (Vide  195, 
Ex.  8  and  10.)  Ans.  15£  19s.  3d.  3|far. 

12.  Add  together  .4  bu.  and  .7  pk. 

Ans.  2  i)k.  2 1  qt. 

13.  Add  together  5.88125  A.  and4  A.  2  R.  35  r. 

Ans.  10  A.  2  R.  16  r. 

14.  Add  together  J  lb.  Troj  and  .583J  oz. 

Ans.  6  oz.  11  dwt.  16  gr. 

15.  Add  together  .875£  and  .75s.        Ans.  18s.  3d. 

SUBTRACTION    OF    COMPOUND    NUMBERS. 

197.    To  subtract  compound  numbers, 

(1.)    Write  the  nitmhers,  as  in  196. 

(2.)  Subtract  the  units  of  the  loioest  value  in  the  sub- 
irahend  from  the  corresponding  units  of  the  7ninuend, 
and  iJ^lace  the  difference  under  the  same  column;  but  if 
the  number  in  the  subtrahend  is  larger  than  that  in  the 
minuend,  add  the  number  standing  over  the  column  to  the 
number  in  the  minuend,  and  subtract  from  the  sum  the 
number  in  the  subtrahend. 

(3.)  If  tfte  jurmher  standii^g  above  any  column  has 
been  employed  in  the  subtraction,  add  1  to  the  units  of 
next  higher  value  in  the  subtrahend ;  after  which  proceed 
exactly  as  with  the  preceding  column,  and  so  on  till  all 
the  columns  have  been  subtracted. 


178  COMPOUND   NUMBERS. 

EXAMPLES. 

1.  From  21  r.  3  ft.  5  in.  take  17  r.  16  ft.  9  in. 


(1.) 

12 

OPERATIONS. 

A7IS. 

(2.) 

12 

21       3 
17     16 

5 
9  .  . 

21 

18 

3 

0 

5 

3 

3r.     24ft. 

Sin. 

3r. 

3ft. 

2. 

Remark. — The  first  result  is  easily  reduced  to  tlie  second  by  ob- 
serving that  6  in.=  ^  ft. 

2.  From  3  fur.  29  r.  2  yd.  1  ft.  take  1  fur.  39  r.  3  yd. 
2  ft.  A71S.  1  fur.  29  r.  4  yd.  6  in. 

3.  From  63  T.  1  hhd.  15  gal.  take  19  T.  3  lihd.  17 
gal.  A71S.  43  T.  1  lilid.  61  gal. 

4.  From  8  bu.  3  pk.  1  qt.  take  3  bu.  2  pk.  7  qt. 

Ans.  5  bu.  2  qt. 

5.  From  25°  4'  27^^  take  17°  20^  40^^ 

A71S.  7°  43'  47'^ 

6.  JV^hat  is  the  difference  between  40  m.  and  39  m. 
7  fur.  39  r.  16  ft.  7  in.  ?  Ans.  1  inch. 

7.  From  77°  0'  15'^  take  71°  3'  30^'. 

Ans.  5°  56'  45''. 

8.  From  85°  30'  take  77°  0'  15".     Ans.  ^  29'  45". 

9.  From  86°  49'  3"  take  79°  55'  38". 

Ans.  6°  53'  25". 

10.  Washington  is  77°  0'  15"  West  Longitude,  and 
Boston  71°  3'  30"  West  Longitude.  What  is  the  differ- 
ence in  the  Longitude  of  these  places  ? 

Ans.  5°  56'  45". 


COMPOUND    NUMBERS.  179 

11.  Louisville  is  85°  30'  W.  L.,  and  Mobile  88°  1'  29" 
W.  L.  What  is  the  difference  in  the  Longitude  of  these 
places  ?  Ans.  2°  31'  29''. 

12.  The  City  of  Mexico  is  in  North  Latitude  19°  25' 
45",  and  Cincinnati  is  in  N.  L.  39°  5'  54".  What  is  the 
difference  ?  Ans.  19°  40'  9". 

13.  The  difference  of  time  between  Greenwich  and 
Milledgeville,  Ga.,  is  5  h.  33  m.  19  sec.  When  it  is 
noon  at  Greenwich,  what  is  the  time  at  Milledgeville  ? 

Ans.  26  m.  41  sec.  past  6  A.  M. 

Remark. — The  time  of  a  place  being  given,  the  time  of  all  places 
east  of  it  is  later,  and  of  all  places  zcest^  earlier. 

198.    To  find  the  time  between  two  dates, 
Write  the  year,  the  order  of  the  month,   (Vide  185, 
Rem.  4,)  and  the  day  of  the  month  of  each  date,  respect- 
ively, under  each  oilier. 
Then  proceed  as  in  197. 


■    ' 

EXAMPLES. 

1.  Find  the 

:,  1867. 

time 

from 

January 

27, 

1865, 

to 

July 

OPERATION. 

12         30 

1867 
1865 

7      4 
1     27 

Ans.  2yr.  5  m.  7  da. 

2.  Find  the  time  from  July  4,  1865,  to  August  1, 
1866.  Ans.  1  yr.  27  da. 

3.  Find  the  time  from  August  1,  1869,  to  September 
9,  1871.  Ans.  2  yr.  1  m.  8  da. 


180  COMPOUND   NUMBERS. 

4.  Find  the  time  from  November  15, 1866,  to  Decem- 
ber 8,  1879.  Ans.  13  yr.  0  m.  23  da. 

5.  Find  the  time  from  January  27,  1866,  to  Septem- 
ber 9,  1871.  Ans.  5  yr.  7  m.  12  da. 

6.  The  Independence  of  the  United  States  was  de- 
clared July  4, 1776.  What  interval  has  passed  on  Jan- 
uary 1,  1867?  A71S.  90  yr.  5  m.  27  da. 

Remark.— The  true  interval  found  by  the  table,  (185,  Rem.  6,) 
is  90  yr.  181  da.) 

7.  A  merchant  bought  at  one  time  8120  gills  of  wine, 
at  another  J  J  of  a  tun,  at  another  |  of  a  hhd.,  and  at 
another  time  .375  of  a  hhd.  What  did  the  whole  cost 
at  11.00  per  gallon  ?  Ans.  $532.00. 

8.  Bought  at  one  time  17  yd.  3  qr.  2  na.  of  broad- 
cloth; at  another  time  lo^%  yd.;  at  another  87.8125 
yd.;  at  another  27  yd.  Ij  qr. ;  and  at  another  time 
29.375  yd.  What  did  the  whole  cost  at  5  J  dollars  per 
yard?  Ans.  ^$965.93 1 . 

9.  From  a  lot  of  land  containing  10  A.  3  R.  ip  r.,  I 
sell  at  one  time  1  A.  2  R.  13  r.;  at  another  time  2  A.  2  R. 
5r.  I  gave  §600.53 J-  for  the  land;  sold  the  first  lot 
at  $70  per  acre,  and  the  second  lot  at  $75  per  acre. 
For  how  much  per  acre  could  I  sell  what  remains,  and 
lose  nothing  ?  Ans.  $44.78. 

10.  Sold  corn  in  three  lots,  viz:.  13  bu.  2  pk.  7  qt. 
1  pt.,  at  60  cts.  per  bu.;  150  bu.  1  pk.  5  qt.,  at  50  cts. 
per  bu. ;  200  bu.  3  pk.  5  qt.  1  pt.,  at  55  cts.  per  bu. 
What  should  I  have  gained  by  selling  all  the  corn  at  56 
cts.  per  bu.  Ans.  $10.48|J. 

11.  A  load  of  hay  weighs  43  cwt.  2  qr.  18  lb.,  includ- 
ing the  wagon  which  weighs  9  cwt.  3  qr.  23  lb.  What 
is  the  hay  worth  at  $7.50  per  ton? 


COMPOUND    NUMBERS.  181 

12.  While  in  London,  I  paid  for  a  vest  1£  13s.  4d.;  a 
coat,  7£  12s.  9d. ;  pants,  2£  3s.  9d. ;  boots  and  hat, 
9£  8s.     How  many  dollars  did  the  whole  come  to  ? 

Am.  $101.115f . 

13.  If  from  a  cask  of  molasses  containing  118  gal., 
20  gal.  1  pt.  leak  out,  for  how  much  must  the  remainder 
be  sold  per  gallon  to  lose  nothing,  the  whole  cost  having 
been  $75?  Ajis.  |0.766. 

MULTIPLICATION   OF   COMPOUND   NUMBERS. 

199.    To  multiply  a  compound  number. 
Multiply  each  of  the  simple  numhei^s  composing  the 
compound  number  hy  the  multiplier^  reducing  lower  to 
higher  units,  as  in  Addition  of  Compound  Numbers. 

EXAMPLES. 

1.  Multiply  3  m.  2  fur.  4  r.  2  ft.  5  in.  by  13. 

OPERATION. 

8  40  16)^  12 


5 
13 


42m.  3fur.  13r.     14A-ft.  5in. 
Ans.  42  m.  3  fur.  13  r.  14  ft.  11  in.,  since  \  ft.=6  in. 

2.  Multiply  12£  4s.  6d.  2far.  by  13. 

Ans.  158£  19s.  2far. 

3.  Multiply  1  lb.  9  oz.  13  dwt.  by  15. 

Ans.  27  lb.  15  dwt. 

4.  Multiply  19  cwt.  3  qr.  23  lb.  by  18. 

Ans.  359  cwt.  2  qr.  14  lb. 


182  COMPOUND    NUMBERS. 

5.  Multiply  18  gal.  3  qt.  1  pt.  by  27. 

Ans.  509  gal.  2  qt.  1  pt. 

6.  Multiply  365  da.  5  h.  48  m.  47.57  sec.  by  7. 

Alls.  2556  da.  16  h.  41  m.  32.99  sec. 

7.  Multiply  4  A.  3  R.  20  r.  4  yd.  7  ft.  47  in.  by  84. 

OPERATION. 

4  40  30)^  9  144 


4 

3 

20 

4 

7 

47 

7 

34 

0 

21 

2f 

6 

il 
12 

409A.  2R.  13r.  lOfyd.  3ft.  60  in. 

Ans.  409A.  2R.  13r.  lly^.  1ft.  24in.,  since  36in.=^ft.  and  2| ft.=Jyd. 

8.  Multiply  4  cwt.  1  qr.  7  lb.  6  oz.  6.5  dr.  by  44. 

Ans.  9  T.  10  cwt.  1  qr.  9  oz.  14  dr. 

9.  Multiply  4  cwt.  18|  lb.  by  476. 

Ans.  99  T.  12  cwt.  6  lb. 

10.  Multiply  21  m.  65  r.  13  ft.  by  5. 

Ans.  106  m.  8  r.  15  ft.  6  in. 

11.  Multiply  1  sq.  r.  57  sq.  ft.  55  sq.  in.  by  7. 

Ans.  8  sq.  r.  129  sq.  ft.  61  sq.  in. 

12.  Multiply  13£  5s.  4.75d.  by  24. 

Ans.  31 8£  9s.  6d. 

13.  Multiply  81£  14s  9d.  by  80. 

Ans.  6539£. 

14.  Multiply  13s.  6d.  Ifar.  by  519. 

Ans.  350£  17s.  3d.  3far. 

15.  Multiply  17  cwt.  3  qr.  10  lb.  by  60. 

Ans.  1071  cwt, 


I 


COMPOUND   NUMBERS.  l«o 

16.  If  a  bale  of  cotton  Is  worth  12£  14s.  6d.  2far., 
what  will  be  the  cost  of  13  bales  at  the  same  rate? 

Ans.  1800.79. 

17.  I  own  16  lots  of  land,  each  containing  5  A.  3  R. 
20  r.     What  is  the  land  worth  at  $40  per  acre  ? 

Ans.  §3760. 

18.  If  a  man  travel  24  m.  4  fur.  4  r.  per  day,  what 
will  it  cost  to  travel  5  days  at  the  rate  of  12  J  cts.  per 
mile  ?  Ans.  |15.32. 

19.  What  will  9  casks  of  sugar  cost  at  $12  per  cwt., 
each  cask  weighing  8  cwt.  2  qr.  12  lb.  ?    Ans.  $930.96. 

20.  If  one  silver  cup  weighs  8  oz.  4  dwt.  10  gr.,  what 
will  6  cups,  each  of  the  same  weight,  be  worth  at  the 
rate  of  $1.25  per  ounce  ?  Ans.  $61.66. 

21.  What  will  9  pieces  of  broadcloth  cost  at  5  dollars 
a  yard,  each  piece  containing  29  yd.  2  qr.  3  na.  ? 

Ans.  $1335.93|. 

22.  A  steamship  in  crossing  the  Atlantic  makes  an 
average  distance  of  250  m.  3  fur.  per  day.  How  far 
will  she  sail  in  9  days?  Ans.  2253  m*  3  fur. 

23.  I  send  to  market  5  casks  of  wine,  each  containing 
123  gal.  2  qt.  1  pt.  What  is  the  wine  worth  at  1  dollar 
per  gallon  ?  Ans.  $618.12  J. 

24.  A  ship  sails  on  the  line  of  the  Equator  5  days, 
at  the  rate  of  2°  15'  20^'  per  day.  How  many  miles 
does  the  ship  make?     (Vide  186,  Rem.  1.) 

Ans.  779.98  miles. 

25.  Suppose  the  ship  in  the  preceding  problem  had 
sailed  the  same  number  of  days  on  a  meridian,  what 
would  have  been  the  distance  made? 

Ans.  778.64  m. 


184  COMPOUND   NUMBERS. 

DIVISION   OF    COMPOUND    NUMBERS. 

200.    To  divide  a  compound  number, 

(1.)  Divide  that  term  of  the  compound  number  which 
has  the  highest  U7iit  value  by  the  divisor^  and  write  the 
quotient  as  the  highest  term  of  the  answer. 

(2.)  Reduce  the  remainder,  if  there  he  any,  to  the  next 
loiver  unit,  adding  that  term  of  the  dividend  which  has 
the  same  unit  value. 

(3.)  Divide  the  sum  as  before,  and  2:)roceed  in  the  same 
manner  till  the  lowest  units  are  reached. 

EXAMPLES. 

1.  Divide  165£  9s.  2  far.  by  13. 

OPERATIONS. 

20       12        4  20     12     4 


13)165        9       0       2         13)165     9     0     2 (12 £  14s.  Gd.  2 far.  .4ns. 

150 

A)is.  12  £  14s.   6d.    2  far.       

9 

Analysis:  165£-f-13=12£  20            Remark. — The  processes  by 

and  9£  Rem.    9£x20-j-9s.  ~~         Short  and  Long  Division  are 

=]89s.    (More  exact,  20s.  -.qo         essentially    the   same.      The 

X9-{-9s.=  189s.)      189s.-r-        operation   by  Long  Division 

13  =  148.  and  7s.  Rem.    7s.  7         is  preferable  till  it  has  become 

Xl2=84d.  _2^        quite  familiar,  since  no  figures 

84d.-i-13— 6d.    and    6d.  34         requiring  attention  are  sup- 

Rem.     6d.X4-f2far.=  26  78         pressed. 

far.     Finally,  26  far.-^13        

=2  far.  5 


26 

2.  Divide  42  m.  3  fur.  13  r.  14  ft.  11  in.  by  13. 

Ans.  199,  Ex.  1. 


COMPOUND   NUMBERS.  185 

3.  Divide  27  lb.  15  clwt.  by  15.     Ans.  199,  Ex.  3. 

4.  Divide  359  cwt.  2  qr.  14  lb.  by  18. 

Ans.  199,  Ex.  4. 

5.  Divide  509  gal.  2  qt.  1  pt.  by  27. 

Ans.  199,  Ex.  5. 

6.  Divide  2556   da.  16  li.  41  m.  32.99  sec.  by  7. 
(Vide  185,  Rem.  2.)  Ans.  199,  Ex.  6. 

7.  Divide  84°  18^  by  15.  Aiis.  5°  37'  12^- 

8.  Divide  83°  19'  45''  by  15.         Aiis.  5°  33'  19". 

9.  Divide  88°  1'  29"  by*'l5.  Ans.  5°  52'  5.9". 

10.  Divide  86°  49'  3"  by  15.        Ans.  5°  47'  16.2". 

11.  If  4  of  a  ship  be  worth  235£  16s.  lid.,  what  is 
the  whole  ship  worth  in  United  States  money? 

A71S.  $7990.46. 

12.  If  I  of  a  ship  be  worth  943£  7s.  8d.-,  what  is  the 
whole  ship  worth  in  francs  ?       Ans.  43485.48  francs. 

13.  If  8  bbl.  of  flour  cost  2£  12s.,  \vhat  will  29  bbl. 
cost?  Ans.  $45,617. 

14.  I  bought  a  tract  of  land  containing  486  A.  2  R. 
30  r.  for  §1000;  subsequently  I  divide  the  land  into  12 
farms,  and  sell  11  of  these  farms  for  |6  an  acre,  reserv- 
ing the  twelfth  as  a  homestead.  How  much  do  I  make 
over  and  above  the  homestead?  A7is.  |1676.78-|. 

15.  From  7  acres  of  land  I  harvest  299  bu.  1  pk.  7  qt. 
of  wheat.  The  tillage  of  the  land  cost  me  $2  an  acre, 
and  I  sell  the  wheat  at  |1.25  per  bushel.  What  do  I 
clear  from  each  acre?  A71S.  $51.47||. 

16.  If  15T.  7  cwt.  2qr.  181b.  of  cotton  cost  $3384.48, 
what  will  1  lb.  cost  ?  What  1  qr.  ?  What  1  cwt.  ?  What 
1  T.  ?  What  would  100  bales  be  worth,  each  bale 
weighing  511  lb.?  Ans.  1  lb.  is  worth  11  cts. 

16 


186  COMPOUND    NUMBERS. 

LONGITUDE    IN    TIME. 

201.  To  cliange  °,  \  ",  of  Longitude,  into  h.  m.  sec. 
of  Time, 

(1.)  Every  point  of  the  Earth's  surface,  except  the 
poles,  moves  through  360°  in  24  hours.     Hence, 
15°  of  Lon.=l  hour  of  Time. 
(2.)  Now  15°=900^  and  lhour==60  minutes.    Hence, 

15'  of  Lon.=l  m.  of  Time. 
(3.)  But  15^:^900^  and  1  m.=60  sec.     Hence, 

15''  of  Lon.=l  sec.  of  Time.     Therefore, 
Divide  the  °,  ',  ",  hy  15,  and  consider  tlie  terms'  of  the 
quotient  as  h.  m.  sec. 

EXAMPLES. 

1.  The  city  of  Lexington  is  84°  18'  W.  L.  When  it 
is  noon  at  Lexthgton,  Avhat  time  is  it  at  Greenwich? 
(Vide  200,  Ex.  7,  and  197,  Ex.  13,  Rem.) 

Ans.  37  m.  12  sec.  past  5  P.  M. 

Remark. — The   Meridian    from    wliicli   Longitude   is    reckoned  • 
passes  through  Greenwich  near  London. 

2.  Milledgeville  is  83°  19'  45"  W.  L.  When  it  is 
noon  at  Greenwich,  what  is  the  time  at  Milledgeville  ? 
(Vide  197,  Ex.  13.)     Ans,  26  m.  41  sec.  past  6  A.  M. 

3.  When  it  is  noon  at  Lexington,  what  time  is  it  at 
Milledgeville?  Ans.  3  m.  53  sec.  P.  M. 

4.  When  it  is  noon  at  Milledgeville,  what  is  the  time 
at  Lexington?        Ans.  11  o'clock  56  m.  7  sec.  A.  M. 

5.  The  city  of  Mobile  is  88°  1'  29"  W.  L.  When  it 
is  noon  at  Mobile,  what  is  the  time  at  Greenwich? 

Ans.  5  o'clock  52  m.  5.9  sec.  P.  M. 


C03IP0UND   NUMBERS.  187 

6.  Louisville  is  85°  30'  W.  L.  What  is  the  difference 
in  time  between  Lexington  and  Louisville  ? 

7.  Washington  is  77°  0'  15''  W.  L.  When  it  is  noon 
at  Washington,  what  is  the  time  at  Louisville  ? 

8.  Boston  is  71°  3'  30''  W.  L.  When  it  is  noon  at 
Washington,  what  is  the  time  at  Boston  ? 

Ans.  2S  m.  47  sec.  P.  M. 

9.  The  Cape  of  Good  Hope  is  18°  29'  E.  L.  When 
it  is  noon  at  Washington,  what  is  the  time  at  Good 
Hope  ?  Ans.  6  o'clock  21  m.  57  sec.  P.  M. 

10.  San  Francisco  is  122°  26'  48"  W.  L.  When  it  is 
8  o'clock  9  m.  47.2  sec.  P.  M.  at  London,  what  is  the 
time'  at  San  Francisco  ?  Ans.  Noon. 

11.  New  York  is  74°  0'  3"  W.  L.  When  it  is  noon 
at  New  York,  what  is  the  time  at  all  the  places  men- 
tioned in  the  preceding  problems  ? 

12.  On  the  morning  of  November  15,  1859,  a  re- 
markable meteor  was  seen  at  New  York,  Albany,  Wash- 
ington, and  Fredericksburg.  At  Washington  the  time 
was  9  o'clock  and  30  m.  What  w^as  the  time  at  the  other 
places,  Albany  being  73°  44'  39"  W.  L.,  and  Fredericks- 
burg 77°  38'  W^  L.  ? 

ANALYSIS    BY    ALIQUOT    PARTS. 

202.    An   aliquot  part  of  any  number  is  an  exact 
half,  third,  fourth,  etc.,  of  the  number.     Thus, 
12J-  cts.  is  an  aliquot  part  of  ^1,  because  12J  is  J-  of  100. 

2*^  is  J  of  4. 

6     is  J  of  24. 

5     is  1  of  20. 

121  is  1  of  25. 


2     R. 

u 

a 

u 

lA, 

u 

6     gr. 

u 

(( 

u 

1  dwt., 

ii 

5     s. 

ii 

u 

u 

1£, 

'i. 

21  lbs. 

u 

u 

« 

1  qr., 

a 

188 


COMPOUND   NUMBERS. 


EXAMPLES. 

-     1.  What  cost  39  A.  2  R.  15  r.  of  land,   at  §87.375 
per  acre. 


OPERATION. 


§87.375 
39 


Therefore, 


Price  of  1  A.  is  .     .     . 

"  39  A.  is  .  . 

"  2  R.  is  .  .  . 

"  10  r.  is .  .  . 

"  5  r.  is    .  .  . 

Price  of  39  A.  2  R.  15  r.  is  |3459.503f  ^  Ans. 

2.  What  cost  39  A.  2  R.  15  r.  of  land,  at  |139.80 
per  acre?  ylii-s.  15535.206 J-. 

3.  What  cost  176  A.  3  R.  25  r.  of  land,  at  §75.375 


3407.625 

43.687.1 

.     J  of  1  A. 

5.460  }i 

.     I  of  2  R. 

2.73011 

.     i  of  10  r. 

per  acre 


Ans.  §13334.308^9. 


4.  What  cost  20  A.  2  R.  24  r.  of  land,  at  §30    per 
acre?  Ans.  §619.50. 

5.  What  cost  10  yd.  3  qr.  2  na.  of  silk,  at  §1.80  per 
yard  ? 

OPERATION. 


Price  of  1  yd.  is 


.80 
10 


Therefore, 


a 

10  yd.  is 

.  18.00 

<£ 

2  qr.  is    . 

.       .90      . 

.  i  of  1  yd. 

u 

1  qr.  is    . 

.       .45     . 

.   i  of  2  qr. 

a 

2  na.  is    . 

.       .225   . 

.  J  of  1  qr. 

Price  of  10  yd.  3  qr.  2  na.  is  §19.575  Ans. 

6.  What  cost  15  yd.  2  qr.  3  na.  of  cloth  at  25  cents 
per  yard?  Ans.  ^S.02^^^. 


COMPOUND    NUMBERS. 


189 


7.  What  cost  25  yd.  1  qr.  3  na.  of  broadcloth,  at  $5.50 
per  yard?  A/i5.  $139.90 1. 

8.  What  cost  67  bu.  3  pk.  7  qt.  of  cranberries,  at  §2 
per  bushel? 

OPERATION. 

Price  of  1  bu.    .     .       §2.00  Therefore, 

67 


"       67  bu.  . 

.     134.00 

^'       2pk.    . 

1.00 

1  bu. 

"       Ipk.    . 

.50 

i  of  2  pk 

"       4  qt.     . 

.25 

i  of  1  pk 

"       2  qt.     . 

.125 

I  of  4  qt. 

"       1  qt.     . 

.0625    . 

\  of  2  qt. 

Price  of  67  bu.  3  pk.  7  qt.  $135.9375  Ans. 

9.  What  cost  125  bu.  3  pk.  1  qt.  of  wheat,  at  87^  cts. 
per  bushel?  Ans.  $110,058. 

10.  What  cost  25  bu.  1  pk.  3  qt.  of   clover  seed,  at 
$5.00  per  bushel?  Ans.  $126.72. 

11.  What  cost  503  bu.  4  qt.  of  corn,  at  43j-  cts.  per 
bushel?.  ^7?s.  $220,117. 

12.  What  cost   76  bu.   1  qt.  of  peas,  at  $1.66|   per 
bushel?  J.ns.  $126.72. 

13.  What  cost  10  bu.  3  pk.  of  apples,  at  50  cts.  per 
bushel?  Ans.  $5.37^. 

14.  What  cost  25  bu.  1  pk.  of  potatoes,  at  35  cts.  per 
bushel?  A71S.  $8.83f. 

15.  What  cost  Ibu.  1  pk.  1  qt.  of  chestnuts,  at  $1.00 
per  bushel?  Ans.  $1.28 J. 

16.  What  cost  17  cwt.  3  qr.   23  lb.  of  hay,  at  $13 
per  ton? 


190 


COMPOUND    NUMBERS. 


OPERATION. 

Price  of 

IT.... 

10  cwt.       .     . 

.      $13.00 

.    therefore, 

iC 

6.50 

.     J  of  1  T. 

u 

5  cwt. 

3.25 

.     i  of  10  cwt 

a 

2  cwt. 

1.30 

J  of  10  cwt. 

a 

2  qr.    . 

.325 

1  of  2  cwt. 

u 

1  qr.    .     . 

.1625     . 

i  of  2  qr. 

a 

5  1b.    . 

.0325 

I  of  1  qr. 

a 

15  1b.    .     . 

.0975     . 

3  times  5  lb. 

a 

lib.    .     . 

.0065     . 

I  of  5  lb. 

Ci 

2  1b.     . 

.013 

2  times  1  lb 

Price  of  17  cwt.  3  qr.  231b.  $11,687    Ans. 

17.  What  cost  3  T.  10  cwt.  3  qr.  of  iron,  at  $30,375 
per  ton?  Ans.  $107.45. 

18.  What  cost  16  boxes  of  sugar,  each  box  contain- 
ing 4  cwt.  3  qr.  18  lb.,  at  $6.65  per  hundredweight? 

Ans.  $524,552. 

19.  What  cost  9  casks  of  sugar,  each  cask  weighing  8 
cwt.  2  qr.  12  lb.,  at  $12  per  cwt.  ?       A7is.  199,  Ex.  19. 

20.  What  cost  16  cwt.  3  qr.  21  lb.  6  oz.  of  rice,  at 
$7.00  per  hundredweight?  Ans.  $118.75. 

21.  What  cost  5  cwt.  2  qr.  of  hay,  at  $27  per  ton? 

Ans.  $7.42f 

22.  What  cost  2  T.  3  cwt.  3  qr.  of  hay,  at  $30  per 
ton?  ^ws.  $65,625. 

23.  What  cost  7  T.  15  cwt.  1  qr.  of  hay,  at  $40  per 
ton?  Ans.  $S10.bO. 

24.  What  cost  the  iron  on  a  track  measuring  3  fur. 
22  r.  3ft.  8 in.,  at  the  rate  of  $4500  per  mile? 


COMPOUND   NUMBERS. 


191 


OPERATION. 


Price  of  1  m. 


§4500.00 


Therefore, 


2  fur.     . 

.      1125.00 

.     J  of  a  mile 

1  fur.     . 

562.50 

.     J  of  2  fur. 

20  r. 

281.25 

.     J  of  1  fur. 

2r. 

28.125 

.  /^  of  20  r. 

3  ft.        . 

2.556  f9- 

.  Jf  of  2  r. 

6  in.       . 

0.426/, 

1  of  3  ft. 

2  in.       . 

0.142  J,  . 

I  of  6  in. 

Price  ofS  fur.  22  r.  3  ft.  8  in.  12000,000     (192-3;  Ex.  40.) 

25.  If  I  ride  5  fur.  3  r.  10  ft.  6  in.  in  a  railroad  car, 
what  ought  to  be  my  exact  fare,  at  the  rate  of  11  cts. 
per  mile?  Ans.  192-3;  Ex.  39. 

26.  What  will  23  gal.  2  qt.  1  pt.  of  wine  cost,  at  §60 
per  hogshead?  Ans.  194-5;  Ex.  25. 

27.  What  cost  11  hhd.  17  gal.  2  qt.  of  wine,  at  |49.77 
per  hogshead?  Ans.  §561.29 J. 

28.  If  one  silver  cup  weighs  8  oz.  4  dwt.  10  gr.,  what 
will  6  cups,  each  of  the  same  weight,  be  worth,  at  the 
rate  of  §1.25  per  ounce?  Ans.  199;  Ex.  20. 

29.  What  will  537 'bushels  of  wheat  cost,  at  §1.374 
per  bushel? 

OPERATION. 


Price   at  §1.00     per  bu. 
"  25  cts.    " 


§537.00 
134.25 


1  of  §1.00. 


121  cts.  " 


67.125  .  h  of  25  cts. 


Price  at  §1.37. 


§738.375  A71S. 


192 


COMPOUND   NUMBERS. 


80.  What  cost  327  bushels  of  potatoes,  at  62^  cts. 
per  bushel?     50  cts.=  i  of  §1;  121  =  i  of  50  cts. 

Ans.  1204.375. 

31.  What  cost  453  bushels  of  corn,  at  87 i  cts.  per 
bushel?      87i'cts.=  50  cts.-{-25  cts.+  12i  cts. 

Ans.  1396.375. 

32.  What  cost  1999  gal.  of  wine,  at  |1.62i  per 
gallon?  JLws.  13248.375. 

33.  What  cost  5794  yd.  of  cloth,  at  |3.16|  per  yard? 
16|  cts.=  i-  of  $1.00.  Ans.  $18347. 66f. 

34.  What  cost  3579  yards  of  cloth,  at  $1.12i-  per 
yard?  $1.18|?  |2.26?  $3.37J?  18|  cts.=3  times  6|  cts. 
=j\  of  $1.00.  First  Ans.  $4026.375. 

35.  What  cost  2468  gal.  of  wine,  at  $1.43}  per 
gallon?  $2.50?  $3.62 J?  $4.56i?  43|-  cts.=:25  cts.+l2^ 
ctB.+6^  cts.  Last  Ans.  $11260.25. 

36.  What  cost  3  T.  10  cwt.  3  qr.  of  iron,  at  6£  4s. 
6d.  per  ton? 

OPERATION. 

Price  of  1  T.      .     .       6£  4s.  6d.         Therefore, 

3 


a 

3T-      . 

.     18    13 

6 

u 

10  cwt.  . 

.       3     2 

3  . 

.    J  of    IT. 

(.6 

2qr.     . 

3 

l/o 

J-^  of  10  cwt 

a 

Iqr.     . 

1 

m 

.  ^i   of    2  qr. 

Priceof3T.  lOcwt.  3qr.22£  Os.  5  J^d-,  or  $106.58. 

37.  What  cost  40  yd.  3  qr.  1  na.  of  broadcloth,  at  1  £ 
per  yard?  Ans.  40 £.  16s.  3d.=-$197.53i. 

38.  What  cost  25  bu.  8  pk.  5  qt.  of  wheat,  at  5  s.  6  d. 
per  bushel  ?  Ans.  7£  2  s.  5 J  ^  d---$34.48l3^.. 


OPERATION. 

3  for  1  jr.  is 

.     .     §325.00 

2 

"      2  yr.  is 

.     .       650.00 

"      3  m.  is 

.     .        81.25 

"    10  da.  is 

.     .           9.02J- 

COMPOLXi)    XUMBEllS.  193 

39  If  I  pay  $325  for  the  use  of  a  certain  sum  of 
money  1  year,  what  ought  I  to  pay  for  the  use  of  the 
same  money  2  yr.  3  m.  10  da.? 


Therefore, 


.     1  of  1  yr. 
.     i  of  3  m. 

Use  for  2  yr.  3  m.  10  da.  is  $740.27-J  Ans.  (185 ;  Rem.  6.) 

40.  At  the  rate  of  $125  per  year,  what  ought  I  to 
pay  for  a  sum  of  money  1  yr.  3.  m.  15  da.  ? 

A71S,  $161.45f. 

41.  At  the  rate  of  SI 75  per  year,  what  ought  I  to 
pay  for  a  sum  of  money  1  yr.  4  m.  21  da.  ? 

A71S.  $243.54^. 

42.  At  the  rate  of  $400  per  year,  what  ought  I  to 
pay  for  a  sum  of  money  1  yr.  5  m.  7  da.  ? 

A71S.  $574.44^, 

43.  At  the  rate  of  $525  per  year,  what  ought  I  to 
pay  for  a  sum  of  money  2  yr.  3  m.  12  da.  ? 

Ans.  $1198.75. 

44.  At  the  rate  of  $1000  per  year,  what  ought  I  to 
pay  for  a  sum  of  money  5  yr.  5  m.  5  da.? 

A71S.  $5430.55|. 

45.  At  the  rate  of  $2000  per  year,  what  ought  I  to 
pay  for  a  sum  of  money  10  yr.  10  m.  10  da.  ? 

Ans.  $21722.22|. 

46.  At  the  rate  of  $750  per  year,  what  ought  I  to 
pay  for  a  sum  of  money  3  yr.  7  m.  19  da.? 

17 


194 


COMPOUND   NUMBERS. 


OrERATION. 

J  for  1  yr.  is     . 

.     .?750.00 
3 

Therefore, 

"       3  yr.  is     . 

.     2250.00 

"       6  m.  is     . 

.       375.00     . 

•    J  of  1  .yr- 

"       1  m.  is     . 

62.50     . 

1  of  6  m. 

"     15  da.  is     . 

31.25     . 

.     I  of  1  m. 

"       3  da.  is     . 

6.25     . 

.     1  of  15  da 

"       1  da.  is     . 

2.08J   . 

.     J  of  3  da. 

Use  for  3  yr.  7  m.  19  da.  |2727.08J   Ans. 

47.  At  the  rate  of  §1000  per  year,  what  ought  I  to 
pay  for  a  sum  of  money  5  yr.  9  m.  27  da.? 

Ans.  S5825.00. 

48.  At  the  rate  of  |2500  per  year,  what  ought  I  to 
pay  for  a  sum  of  money  4  yr.  11  m.  25  da.  ? 

Am.  §12465.27|. 

49.  At  the  rate  of  §825  per  year,  what  ought  I  to 
pay  for  a  sum  of  money  6  yr.  2  m.  18  da.? 

Ans.  §5128.75. 

50.  The  rent  of  a  farm  is  §1150  per  year.     What 
ought  to  be  paid  during  an  occupancy  of  7  yr.  7  m.? 

Ans.  §8720.831 . 

51.  What  ought  I  to  pay  for  the  use  of  §5000  during 
4  yr.  3  m.  20  da.,  at  the  rate  of  §400  per  year? 

J.7is..§1722.22|. 

52.  What  ought  I  to  pay  for  §1850  during  3  m.  3  da., 
at  the  rate  of  §148  per  year?  Ans.  §38.23 1. 

53.  What  ought  I  to  pay  for  §9215  during  3  m.  3  da., 
at  the  rate  of  §737.20  per  year?  Ans.  §190.44 J. 

54.  What  ought  I  to  pay  for  the  use  of  §8000  from 


COMPOUND   NUMBERS.  195 

January  27,  1866,  to  July  4,  1868,  at  the  rate  of  $640 
per  year?— (198;  Ex.  1.)  Ans.  $1559.111. 

55.  What  ought  I  to  pay  for  the  use  of  $7250  from 
July  4,  1865,  to  August  1,  1866,  at  the  rate  of  $580 
per  year?  Ans.  $623.50. 

56.  What  ought  I  to  pay  for  the  use  of  $5280  from 
August  1,  1869,  to  September  9,  1871,  at  the  rate  of 
$422.40  per  year?  Ans.  889.38|. 

57.  If  I  pay  12  cts.  per  year  for  the  use  of  $1,  "what 
ought  I  to  pay  for  its  use  3  yr.  5  m.  15  da.  ? 

Ans.  $0,415. 

58.  If  I  pay  $99.75  per  year  for  the  use  of  $950, 
what  ought  I  to  pay  for  the  use  of  the  same  sum  2  yr. 
4m.  20  da.?  Ans.  $238.29^. 

59.  What  ought  I  to  pay  for  the  use  of  $421.40  for 
3  yr.  5  m.  15  da.,  at  the  rate  of  $25,284  per  year? 

OPERATION. 

Use  for  1  yr.  is     .         .     $25,284 

3 


a 

3  yr.  is      . 

75.852 

a 

4  m.  is 

8.428 

.     1  of  1  yr. 

a 

1  m.  is 

2.107 

.     1  of  4  m. 

u 

15  da.  is 

.     ,    1.0535 

.     1  of  1  m. 

Use  for  3  yr.  5  m.  15  da.       $87.4405 

203.   Review  in  Addition. 

1.  Find  the  value  of  123+45+2004.       Ans.  2172. 

2.  Find  the  value  of  21+105+710.  Ans.  836. 

3.  Find  the  value  of  7+90+1041.  Aiis.  1138. 


196  COMPOUND    NUMBERS. 

4.  Find  the  value  of  50+75+432ia   ^m  43335. 

5.  Find  the  value  of  ?1 2 3+^45+12 004. 

Ans.  $2172. 

6.  Find  the  value  of  21  £+105 £+710 £. 

Ans.  836  £. 

7.  Find  the  value  of  7  m.+90  m.+1041  m. 

Ans.  1138  ra. 

8.  Find  the  value  of  50  pk.+75  qt.+43210  pt. 

Ans.  44160  pt. 

9.  Find  the  value  of  .123+.45+.2004.   Ans.  .7734. 

10.  Find  the  value  of  .21+.105+.71.    J.ns.  1.025. 

11.  Find  the  value  of  .7+.9+.1041.    Ans.  1.7041. 

12.  Find  the  value  of  .5+.75+.4321.  Ans.  1.6821. 

13.  Find  the  value  of  $0.123+S0.45+S0.2004. 

Ans.  $0.7734. 

14.  Find  the  value  of  .21  £+.105 £+.71  £. 

Ans.  1.025  £. 

15.  Find  the  value  of  .7  1.+.9  m.+  .1041  fur. 

Ans.  24.1041  fur. 

16.  Find  the  value  of  .5  pk.+.75  qt.+.4321  pt. 

A71S.  9.9321  pt. 

17.  Find  the  value  of  1.23+4.5+20.04. 

Ans.  25.77. 

18.  Find  the  value  of  2.1+1.05+7.1.    Ans.  10.25. 

19.  Find  the  value  of  7+90+10.41.    Ans.  107.41. 

20.  Find  the  value  of  50+7.5+.4321. 

Ans.  57.9321. 

21.  Find  the  value  of  |1.23+$4.50+|20.04. 

A71S.  $25.77. 

22.  Find  the  value  of  2.1  £+1.05 £+7.1  £. 

Ans.  10.25  £. 


COMPOUND    NUMBERS.  197 

23.  Find  the  value  of  7  l.-f9  m.+10.41  fur. 

Ans.  250.41  fur. 

24.  Find  the  value  of  50  pk.+7.5  qt.+.4321  pt. 

A71S.  815.4321  pt. 

25.  Find  the  value  of  |+i+|.  ^ns.  IJ. 

26.  Find  the  value  of  J  +  |  +  |.  Ans.  If  §. 

27.  Find  the  value  of  ^^-{-f^-\-^^.  Ans.  l^gf . 

28.  Find  the  value  of  ii-hyVs  +  Z/s-  -^ns.  f  i§. ' 

29.  Find  the  value  of  ||+li+li.  Ans.  |1.12i. 

30.  Find  the  value  of  i  £+|  s.-[-|  d. 

Ans.  128J  d. 

31.  Find  the  value  of  y^^  1.+/^-  m.+  ^g  fur. 

Ans.  lOif  fur. 

32.  Find  the  value  of  ^|  pk.+  fVs  ^t-+i¥3  P^- 

Ans.  4||  pt. 

33.  Find  the  value  of  123|+45|+2004|. 

Ans.  2173|. 

34.  Find  the  value  of  21i+105H-710|. 

Ans.  837|g. 

35.  Find  the  value  of  7i\+90/o+1041/5. 

A71S.  1139^i§. 

36.  Find  the  value  of  50^J+75/5V+43210//3. 

Ans.  43335|i§. 

37.  Find  the  value  of  1.231+4.5-1+20.041. 

Ans.  25.791. 

38.  Find  the  value  of  2.1i+1.05|+7.10|. 

Ans.  10.31/5. 

39.  Find  the  value  of  .7f,-+9.0/3+10.41/5. 

^   Ans.  20.18H§. 

40.  Find  the  value  of  5.0iJ+7.5jV5+4321.0//3. 

Ans.  4333.56.\. 


198  COMPOUND   NUMEERi. 

41.  Find  the  value  of  $1.23|+§4.51i+§20.04|. 

A71S.  $25,791. 

42.  Find  the  value  of  2.15X+1.05|£+T.108£. 

A71S.  10.3141  £. 

43.  Find  the  value  of  .7i\  r.+9.0/^  ft.H- J  in. 

Ans.  1  r.  4  ft.  7  in. 

44.  Find  the  value  of  5.0^|  A.+7.5jV5  R.+l  r. 

Ans.  6  A.  3  R.  21.48  r. 

45.  Add  together  l£  2s.  6d.;  9£  8s.  9d.;  12£  13s. 
2d.;  and  4£  7s.  3d.  Ans.  27£  lis.  8d. 

46.  Add  together  15  r.  16  ft.  6  in.;  33  r.  14  ft.  7  in.; 
and  19  r.  8  ft.  9  in.  Ans.  69  r.  6  ft.  10  in. 

47.  Add  together  5  A.  2  R.  7  r. ;  8  A.  3  R.  32  r. ;  and 
9  A.  1  R.  27  r.  Ans.  23  A.  3  R.  26  r. 

48.  Add  together  3  hhd.  62  gal.  3  qt.;  2  hhd.  16  gal. 
1  qt.;  3  hhd.  57  gal.  2  qt.;  3  hhd.  45  gal.  3  qt.;  2  hhd. 
59  gal.  3  qt. ;  and  3  hhd.  39  gal.  2  qt. 

Ans.  20  hhd.  29  gal.  2  qt. 

204.   Review  in  Subtraction. 

1.  From  1001  take  763.  Ans.  238. 

2.  From  3999  take  455.  Ans.  3544. 

3.  From  31015  take  9999.  Ans.  21016. 

4.  From  11111  take  7778.  Ans.  3333. 

5.  From  $245  take  $78.  Ans.  $167. 

6.  From  24  pears  take  19  pears.  Ans.  5  pears. 

7.  From  91  apples  take  74  apples.  Ans.  17  apples. 

8.  From  375  da.  take  109  da.  Ans.  266  days. 

9.  From  .763  take  .1001.  Ans.  .6629. 

10.  From  .455  take  .3999.  Ans.  .0551. 

11.  From  .45  take  .0073.  Ans.  .4427. 


COMPOUND   NUMBERS.  199 

12.  From  .39  take  .039.  Ans.  .351. 

13.  From  §0.37  take  |0.227.  Ans.  |0.143. 

14.  From  .5  pears  take  .25  pears.    Ans.  .25  pears. 

15.  From  .91  apples  take  .74  apples. 

Ans.  .17  apples. 

16.  From  .67  lb.  take  .4937  lb.  Ans.  .1763  lb. 

17.  From  10.01  take  7.63.  Ans.  2.38. 

18.  From  39.99  take  .455.  Ans.  39.535. 

19.  From  3.8  take  1.005.  Ans.  2.795. 

20.  From  37  take  19.04.  Ans.  17.96. 

21.  From  |3.25  take  $2.12.  Ans.  |1.13. 

22.  From  3.5  da,  take  2.85  da.  Ans.  .65  days. 

23.  From  43.7  m.  take  27.43  m.         Ans.  16.27  m.  ' 

24.  From  9.5  sec.  take  4.67  sec.         Ans.  4.83  sec. 

25.  Find  the  value  of  3— ^  ;  of  4— |.  Ans.  2^  ;  3}. 

26.  Find  the  value  of  |— | ;  of  |— ^.       Ans.  | ;  |. 

27.  Find  the  value  of  3^—2 J  ;  of  4i— 3l. 

Ans.  1|;  IgL. 

28.  Find  the  value  of  8— 3^  ;  of  12—41. 

Ans.  4i  ;  7|. 

29.  Find  the  value  of  3.8— 2j  ;  of  37— 19-^^ 

Ans.  1.3 ;  17.96. 

30.  Find  the  value  of  4.3^—2.331;  of  84— 3.14f. 

Ans.  2.01f ;  5. 

31.  Findthe  value  of  42— .OOOJ;  37—4.000'". 

Ans.  41.9991;  32.999 J. 

32.  Find  the  value  of  860.4581— 25.1  J. 

Ans.  835.3248f . 

33.  From  |  h.  take  i  m.  Ans.  29  m.  45  sec. 

34.  From  i  da.  take  J  h.  Ans.  5.  h.  45  m. 

35.  From  |  m.  take  *  in.    Ans.  191  r.  16  ft.  5J  in. 


200 


COMPOUND   NUMBERS. 


36.  From 


J  A.  take  J 


sq.  m. 


».75i. 


A71S.  19r.  272  ft.  35f  in. 
Ans.  $0,495/5. 
Ans.  7  ft.  8|g  in. 
Alts.  2  gal.  -i  qt. 

A718.  |1.00/g. 

Ans.  $12.48. 


37.  From  §1.25|  take 

38.  From  8.3  J  ft.  take  7.0|  in. 

39.  From  2J  gal.  take  J  qt. 

40.  From  7i  francs  take  371  cts. 

41.  From  3|£  take  3}  dollars. 

42.  From  4i  bu.  take  3j  pk.  J.ws.  3  bu.  2  pk.  4  qt. 

43.  From  9  m.  take  8  m.  7  fur.  39  r.  16  ft.  5  in. 

Ans.  1  inch. 

44.  From  8  m.  7  fur.  16  ft.  5  in.  take  5  m.  6  yd.  11  in. 

A71S.  3  m.  6  fur.  39  r.  14  ft. 

45.  From  1£  21s.  13d.  5 far.  take  5s.  3d.  7 far. 

Ans.  l£16s.  9d.  2  far. 

46.  From  9  bu.  3  pk.  5  qt.  take  4  bu.  7  pk.  15  qt. 

Ans.  3  bu.  2  pk.  6  qt. 

I305.  Review  in  Multiplication  and  Division. 


1.  Multiply  45  by  45. 

3.  Multiply  101  by  1001. 

5.  Multiply  .45  by  .45. 

7.  Multiply  .101  by  .1001. 

9.  Multiply  4.5  by  45. 
11.  Multiply  .0009  by  .0009. 
13.  Multiply  $45  by  75. 
15.  Multiply  365  da.  by  17. 
17.  Multiply  375.61b.  by  .125. 
19.  Multiply  03.5  bu.  by  .78. 
21.  Multiply  74  by  ^f-y. 
23.  Multiply  vjVj  by  370. 
25.  Multiply  H  by  ff. 
27.  Multiply  4  by  zll. 
29.  Multiply  6^|  by  24. 
31.  Multiply  4^  by  3f 

Am.  145. 


2.  Divide  2025  by  45. 

4.  Divide  101101  by  1001. 

6.  Divide  .2025  by  .45. 

8.  Divide  .0101101  by  .1001. 
10.  Divide  202.5  by  4.5. 
12.  Divide  .00000081  by  .0009. 
14.  Divide  $3375  by  75. 
10.  Divide  6205  da.  by  17. 
18.  Divide  4G.95  by  .125. 
20.  Divide  49.53  bu.  by  .78. 
22.  Divide  6  by  ^i^^. 
24.  Divide  i||o  by -870. 
20.  Divide  j%  by  j^^. 
28.  Divide  14  by  3i|.   , 
30.  Divide  7|  by  3. 
32.  Divide  14^  by  S^V 

Ans.   415. 


{J03il'0UXD   ^' UMBERS.  201 


33.  Multiply  .OOOi  by  4.8. 
85.  Multiply  S.Of  by  2.00i. 
37.  Multiply  375f  r.  by  |. 
39.  Multiply  63^  A.  by  |f 
41.  Multiply  3.75ir.  by  51. 
43.  Multiply  8.61  bu.  by  129. 


31.  Divide  .0016  by  .0001. 

36.  Divide  6.031 1|  by  2.00^. 

38.  Divide  46 |i  r.  by  i. 

40.  Divide  49.53  A.  by  ff. 

42.  Divide  191f  i  by  51. 

44.  Divide  1112f  bu.  by  8|. 


45.  Multiply  26  £  14  s.  8d.  3  far.  by  11. 

Ans.  294:  £  2  s.  Od.  Ifar. 

46.  Multiply  1  m.  1  fur.  1  r.  1  ft.  1  in.  by  27. 

Ans.  30  m.  3  fur.  28  r.  12  ft.  9  in. 

47.  Multiply  8  T.  1  cwt.  3  qr.  7  lb.  5  oz.  6  dr.  by  3^ 

Ans.  28  T.  6  cwt.  1  qr.  13  lb.  2  oz.  13  dr. 

48.  Multiply  365  da.  5  li.  48  m.  47.57  sec.  by  6. 

Ans.  2191  da.  10  h.  52  m.  45!'42  sec. 

49.  Add  together  10  apples,  4  pears,  and  6  peaches 
(Vide  171.) 

50.  Add  together  3  ft.  and  4  in.       Ans.  40  inches. 

51.  From  §100  take  4  bu.  2  pk.  1  qt  The  problem 
is  absurd.     Why? 

52.  I  buy  4  bu.  2  pk.  1  qt,  of  cherries  at  50  cts.  per 
qt.,  and  pay  for  them  out  of  a  §100  bill.  How  much 
money  do  I  receive  in  change?  Aiis.  §27.50. 

53.  Multiply  25  cts.  by  25  cts.— (Vide  82.) 

54.  Divide  25  rails  by  25  rails.— (Vide  96.) 

55.  Reduce  each  of  the  following  expressions  to  a 
compound  number:  3.23125 £;  S^Vo^;  3.23|£;  64.- 
625s.;  64gs;  64.6is.;  775 id.;  775.5 d.;  3102 far. 

Ans.  3£4s.  7d.  2 far. 

56.  Reduce  1  yr.  2  mo.  6  da.  to  months  and  decimals 
of  a  month.  1  yr.  3  mo.  9  da.;  2  yr.  4  mo.  12  da.;  and 
3  yr.  6  mo.  15  da. 

J.ns.  14.2  mo. ;  15.3  mo.;  28.4mo.;  42.5  mo. 


202  PERCENTAGE. 

PERCENTAGE. 


206.  Percentage  embraces  those  operations  in 
which  numbers  are  compared  with  100  as  a  unit. 

SOT.  Per  centum,  in  Latin,  signifies  by  the  hundred, 
but  the  contraction  per  cent,  is  used,  for  the  most  part, 
as  a  synonym  of  the  word  hundredth  or  hundredths. 
Thus, 

1  per  cent,  of  25  is  the  same  as  1  hundredth  of  25. 

2  per  cent,  of  50  is  the  same  as  2  hundredths  of  50. 
25  per  cent,  of  a  number  is  the  same  as  25  hundredths 
of  it. 

The  sign  ^  is  the  same  as  the  contraction  per  cent. 

The  RATE  PER  CENT,  is  the  number  of  hundredths. 
Thus, 

■  5  ^  of  400  is  read  5  per  cent,  of  400,  and  has  the 
same  meaning  as  5  hundredths  of  400,  that  is,  yj^  of 
400,  or  .05  of  400. 

7  %  of  a  number  is  7  hundredths  of  it. 

1%  of  a  number  is  f  hundredths  of  it,  that  is  ^f^ 
of  it. 

^iOS.  To  represent  decimally  any  given  rate  per 
cent., 

Write  the  given  rate  as  so  many  hundredths. 

EXAMPLES. 

1.  Represent  the  following  rates  per  cent,  decimally, 
viz:  1%;  2%;  12%;  10%;  25%;  47%;  100%; 
125%;  200%;  1000%;  1275%. 

Ans..O\]  .02;  .12;  .10;  .25;  .47;  1.00;  1.25;  2.00;  10.00;    12.75. 


TERCENTAGE.  203 

2.  Represent  decimally  h%;i%;  ~l%;  Jo%>  i  o  %  ; 

A71S.  .005;  .0025;  .001;  .0005;  .001;  .0002;  .0075;  .004. 

3.  Represent  decimally  1^^;  2\%;  3j%;  lj^%; 
^i\fo;  11%;  H%;  H%. 

Ans.  .015;  .0225;  .032;  .0105;  .061;  .075;  .051;  .041. 

4.  Represent  decimally  ^^-^  ;  ^%%;  |%;  125|%; 
4Jg%.  ^7is.  .00025;  .00075;  .00375  ;  1.25|;  .040625. 

209.  To  find  what  rate  per  cent,  a  given  decimal 
represents, 

Multiply  the  decimal  hy  100  and  reduce  the  result^  if 
necessary^  as  in  164. 

EXAMPLES.     . 

1.  What  rate  per  cent,  does  .01  represent?  .04?  .05? 
.07?  .1?  .2?    Ans,l%',4.%',  5%;  1  %;  10%;  20%. 

2.  What  rate  per  cent,  does  .00428f  represent?  .000|? 
.008 1  ?     (Vide  164,  Ex.  7.)        Ans,  |  %;  -^^%',  %%. 

3.  What  rate  per  cent,  does  .001  represent?  .0075? 
.041^?  1.00?  Ans,  ^\%;  f  %;  4i%;  100%. 

4.  What  rate  ^er  cent,  does  .007  represent?  .0007? 
1.07?  Ans.  /^%;  tJo%;  107%. 

!S10.  To  find  the  per  cent,  (or  percentage)  of  a  num- 
ber at  a  given  rate, 

Multiply  the  given  number  hy  the  rate  per  cent,  ivrit- 
ten  as  a  decimal.  The  product  will  be  the  per  cent. 
required. 

EXAMPLES. 

1.  Find  3%  of  25;  -J%  of  |500 ;  and  1.^^%  of 
7000  lbs.  of  coff'ee. 


204  PERCENTAGE. 

OPERATIONS. 

25  $500  7000  lb. 

.03  .005  .0105 


.75=1  Ans.        12.500  Ans.        73.5000=-73i  lb.    Ans. 

2.  Find  1  %  of  $120  ;  2  %  of  $410  ;  7  %  of  140  bu. ; 
9%  of  555  da.     (211,  Ex.  2.) 

3.  Find  6%  of  333  fur.;  5%  of  400  m.;  1%  of 
$1000;  J  %  of  $3333. 

4.  Find  1%  of  $7000;  n%  of  $9000;  2^%  of  $700; 
J^%  of  $10000. 

5.  Find  2%  of$125;7^;ll%;25%;75%;100%; 
150%. 

6.  Find  11%  of  Gets.;  7cts.;  $1.20;  $1.75;  $3.50; 
$7.62i;  $9.45. 

7.  Find  25%  of  $25;  33i  %  of  $500;  16f  %  of 
$8000. 

8.  Find  101 J  %  of  $505;  202  J  %  of  $404;  1000% 
of  $23.10. 

9.  Received  at  Mobile  from  New  Orleans  200  lihds. 
of  sugar ;  but  in  discharging  the  cargo  2  %  of  the  sugar 
was  lost.     How  much  remains  to  be  sold? 

Ans.  196  hhds. 

10.  The  steamer  Indiana  started  from  Vicksburg  with 
500  bales  of  cotton  on  board.  On  the  trip  to  New  Or- 
leans 14%  of  the  cotton  was  transferred  to  another  boat. 
How  much  remained  on  board?  Ans.  430  bales. 

11.  Bought  at  New  Orleans  75  hhds.  of  molasses,  but 
on  its  arrival  at  Cincinnati  4%  is  missing.  How  much 
remains  to  be  sold?  Ans.  72.  hhds. 

12.  Shipped  at  Havana  for  New  York  1250  boxes  of 


PERCENTAGE.  205 

oranges;  on  the  passage  14  fo  of  the  oranges  Avere 
thrown  overboard.  How  many  boxes  arrived  in  New 
York?  Ans.  1075  boxes. 

13.  On  a  trip  from  New  Orleans  to  New  York  I  ex- 
pend 25%  of  my  money.  I  started  with. §360.  How 
much  did  I  have  on  arriving  at  New  York?  (Vide  167, 
Ex.  103.)  Ans.  §270. 

14.  During  a  storm  a  captain  threw  overboard  16|  ^ 
x)f  a  cargo  of  cotton.  He  left  New  Orleans  with  720 
bales;  with  how  many  did  he  arrive  at  Liverpool^  (167, 
Ex.  104.)  Ans.  600  bales. 

15.  My  wine  made  during  the  year  1865  was  378 
gallons.  Reserving  7^%,  I  sell  the  remainder  at  §4.50 
per  gallon.  What  did  the  crop  bring  me?  (167,  Ex. 
105.)  Ans.  §1579.50. 

16.  My  wine  made  during  the  year  1866  w^as  450 
gallons.  Reserving  10  ^  for  private  use,  I  sell  the  re- 
mainder at  §3.75  per  gallon.  Which  year  was  most 
profitable,  1865  or  1866,  and  by  how  much? 

Ans.  1865,  by  §60.75. 

17.  I  buy  sugar  for  §1700,  and  sell  so  as  to  clear  5  ^ 
on  the  cost.     How  much  do  I  get?  Ans.  §1785. 

18.  A  merchant  failing,  pays  his  creditors  30^.  He 
owes  A  §2500;  B  §4000;  and  C  §4500.  What  will  each 
receive?  Ans.  A  §750;  B  §1200;  and  C  §1350. 

19.  I  pay  10%  of  my  salary  for  board;  i%  for 
washing ;  l2  %  for  clothes,  and  8  %  for  other  expenses. 
How  much  do  I  clear  from  a  salary  of  §2000  ? 

Ans.  §1395. 
Sll.  To  find  the  rate  per  cent,   of  a  number  at  a 
given  percentage, 


'    c 

} 

206  PEllCEXTAGE. 

Divide  the  given  percentage  hy  the  number  of  which 
the  rate  per  cent,  is  required.  The  quotient  will  repre- 
sent the  rate  per  cent,  decimally,  and  may  be  changed 
as  in  209. 

EXAMPLES. 

1.  What  rate  %  of  25  is  .75?  of  $500  is  $2.50?  of 
7000  lb.  of  coffee  is  73. \  lb.  of  coffee? 


OPERATIONS. 

(1.)  "  (2.)  (3.) 

25).75  500)2.50  7000)73.5 

.03=%  Ans.     .005= J  %  Aiis.   .0105=1  ^.^^^    -^ns. 

2.  What  rate  %  of  $120  is  $1.20?  of  $410  is  $8.20? 
of  140  bu.  is  9|  bu.  ?  of  555  da.  is  49  ^  §  da.  ? 

3.  What  rate  %  of  333  fur.  is  19|~§fur.?  of  400  m. 
is  20  m.?  of  $1000  is  $5.?  of  $3333  is  $11.11? 

4.  What  rate  %  of  $7000  is  $17.50?  of  $9000  is 
$135?  of  $700  is  $15f  ?  of  $10000  is  $5? 

5.  What  rate  %  of  $125  is  $2.50?  $8.75?  $13.75? 
$31.25?  $93.75?  $125?  $187.50? 

6.  What  rate  %  of  6  cts.  is  j%  m.  ?  of  7  cts.  is  1  j^  m.? 
of  $1.20  is  If  cts.?  of  $1.75  is  2|  cts.?  of  $3.50  Is  5J- 
cts.?  of  $7.62i  is  11/5  cts.?  of  $9.45  is  14/^ cts.? 

7.  What  rate  %  of  $25  is  $6 J?  of  $500  is  $166|? 
of  $8000  is  $1333|  ? 

8.  What  rate  %  of  $505  is  $511.06?  of  $404  is 
$818.10?  of  $23.10  is  $231? 

9.  From  a  ship  having  on  board  200  bales  of  cotton, 
4  bales  fell  into  the  sea,  and  wer^  lost.  What  rate  % 
were  these  four  bnles  of  the  whole  number  on  board? 


rEllCEXTAGE.  207 

10.  A  boy  commenced  play  with  200  marbles,  and 
ended  with  196.     What  was  his  rate  ^  of  loss? 

11.  A  ship  sailed  from  New  Orleans  for  Liverpool 
with  a  cargo  of  500  bales  of  cotton.  The  ship  reached 
her  port  with  only  430  bales,  the  remainder  "having  been 
thrown  overboard.  With  what  rate  %  of  her  cargo  did 
she  reach  Liverpool,  and  what  rate  %  had  been  thrown 
into  the  sea? 

12.  Out  of  1250  boxes  of  oranges  shipped  at  Havana, 
1075  arrived  in  New  York.  What  rate  %  of  the  cargo 
had  been  lost  ? 

13.  I  started  from  Charleston  with  $360,  and  arrived 
in  Quebec  with  §270.  What  rate  %  of  my  money  had 
been  expended  in  the  trip? 

14.  Out  of  my  wine  made  in  1865,  which  was  378 
gallons,  I  sold  a  quantity  amounting,  at  the  rate  of  ?4j 
per  gallon,  to  $1579|,  having  reserved  the  remainder 
for  private  usfe.  What  rate  ^  on  the  whole  wine  was 
reserved  ? 

15..  My  wine  made  in  1866  had  increased  at  the  rate 
of  19 2^-  ^/o  on  that  made  in  1865,  but  tHe  market  being 
dull  I  sold  a  portion  of  it  for  §3.75  per  gallon,  clearing 
§60.75  less  than  on  the  previous  year.  What  rate  ^  of 
the  wine  of  1866  remained  unsold? 

!S1!^.  To  find  a  number  on  which,  at  a  given  rate  ^, 
a  given  percentage  may  be  obtained. 

Divide  the  given  'percentage  hy  the  given  rate  per  cent, 
expressed  decimally.  .The  quotient  Avill  be  the  required 
number. 


208  ^  PERCENTAGE. 

EXAMPLES. 

1.  What  number  is  that  of  which  3%  is  .75?  1%  of 
how  many  dollars  is  $2.50  ?  l^^  %  of  how  many  pounds 
is  73i  pounds? 

OPERATIONS. 

(1.)  (2.)  (3.) 

.03).75  .005)12.500  .0105)73.5000  lb. 

25  Ans.  1500  Ans.  7000  lb.  Ans. 

2.  8^  of  what  number  will  produce  125? 

Am.  1562.50. 

3.  8^  of  how  many  dollars  will  produce  |175? 

A71S.  12187.50. 

4.  5  ^  of  how  many  dollars  will  produce  |400  ? 

Ans.  18000. 

5.  4^  of  how  many  dollars  will  produce  §250? 

Ans.  16250. 

6.  On  a  pleasure  excursion  I  spend  $90,  which  I  find 
is  25  fo  of  the  money  with  which  I  started.  How  much 
money  have  I  still  on  hand?  Ans.  (211,  Ex.  13.) 

7.  If  I  pay  8  ^  of  a  sum  of  money  for  its  use  during 
a  year,  and  thereby  pay  |525,  what  amount  of  money 
do  I  have  the  use  of?  Ans.  $6562.50. 

8.  If  I  pay  6^  of  Sb  sum  of  money  for  its  use  during 
a  year,  and  thereby  pay  $750,  what  amount  of  money 
do  I  have  the  use  of?  Ans.  $12500. 

9.  I  borrow  $350  for  1  year,  and  agree  to  pay  7% 
of  the  sum  for  its  use.     How  much  do  I  pay  ? 

Ans.  $24.50. 

10.  If  I  pay  $24.50  for  the  use  of  $350  for  1  year, 
what  rate  %  on  the  money  do  I  pay?      *      Ans.  7%, 


PERCENTAGE.  209 

11.  What  amount  of  money  can  I  get  the  use  of  for 
1  year  by  paying  $24.50,  that  being  7  ^  on  the  money 
borrowed?  Ans.  $350. 

S13.  A  number  being  given  which  is  a  given  rate 
per  cent,  more  than  another  number,  to  find  that  other 
number, 

Divide  the  given  number  by  \  -\-  the  rate  per  cent, 
written  as  a  decimal. 

EXAMPLES. 

1.  560  is  12  fo  more  than  a  certain  number.  What 
is  that  number? 

Remark. — Thp  question  is  precisely  the  operation. 

same  as  this;  560  is  iif  of  what  number? 
and  (560-112)  X  100=560-1.12.  1.12)560.00 

500  Ans. 

2.  1000  is  33|  fo  more  than  a  certain  number.  What 
is  that  number?     1000-^1.33|=|  of  1000. 

Ans.  750. 

3.  $150  is  20%  more  than  what  sum?     Ans.  $125. 

4.  $140  is  16?  fo  more  than  what  sum? 

Ans.  $120. 

214.  A  number  being  given  which  is  a  given  rate  p'er 
cent,  less  than  another  number,  to  find  that  other  number. 

Divide  the  given  number  by  X  —  the  rate  per  cent,  ivr it- 
ten  as  a  decimal. 

EXAMPLES. 

1.    270    is   25  %   less   than   what         operation. 

number?  1-^9 

Remark. — The  question   is   the   same  as  

this:  270  is  ^^-^  of  what  number?  and  (270       •'^5)270.00 
-f-75)  X  100=270— 75.      (Vide  167.  Ex.  98.)  36O  im 

18 


210  PERCENTAGE. 

2.  $1.40  is  30%  less  than  what  sum? 

Ans.  $2.00. 

3.  $4.50  is  25%  less  than  what  sum? 

A71S.  $6.00 

4.  $8.75  is  33J%  less  than  what  sum?     (|  of  8.75.) 

Ans.  $13.12J. 

APPLICATIONS   OF   PERCENTAGE. 

215.  Insurance  is  a  contract  made  between  parties, 
by  which  the  one  binds  itself,  for  a  consideration,  to  re- 
imburse the  other  for  losses  of  property  occasioned  by 
fires,  or  other  casualties. 

(1.)  The  party  taking  the  risk  is  called  the  Under- 
writer. 

(2.)  The  party  protected  is  called  the  Insured. 

(3.)  The  Policy  is  the  written  contract  of  insurance. 

(4.)  The  Premium  is  the  sum  paid  for  insurance. 

(5.)  The  premium  is  usually  a  percentage  on  the  value 
of  the  property  insured,  and  is  paid  at  the  time  the  pol- 
icy is  drawn. 

'  310.    To  find  the  Premium,  when  the  amount  insured 
and  the  rate  per  cent,  of  insurance  are  given. 

Multiply  the  amount  insured  hy  the  rate  %,  wiHtten 
decimally. — (Vide  210.) 

EXAMPLES. 

1.  What  premium  must  be  paid  annually  for  insuring 
a  house  worth  $4500,  at  -J  %  ?  Ans.  $11.25. 

2.  What  is  the  premium  on  a  cargo  of  cotton  valued 
at  $2500,  at  1  %  ?  Ans.  $3,125. 


PEllCE^'TAGE.  211 

3.  What  is  the  premium  on  a  cargo  of  goods  carried 
from  New  York  to  Mobile,  the  goods  being  valued  at 
112500,  and  insured  at  Ij  %  ?  Ans.  $187.50. 

4.  I  have  a  house  worth  §4500,  and  insure  it  for  |  of 
its  value  at  1|%  per  annum.  What  is  the  expense  of 
insurance,  including  $2.00  for  the  policy? 

Ans.  §50.00. 

5.  I  lose  by  fire  four  houses,  valued  at  §25000.  On 
this  property  I  had  paid  annually,  during  five  years,  a 
premium  of  lj%  on  the  entire  value.  The  policy  be- 
ing good,  what  have  I  saved  by  insuring  the  property? 

Ans.  §28437.50. 

217.  To  find  for  what  sum  a  policy  must  be  taken 
out,  at  a  given  rate  per  cent.,  to  cover  both  property  and 
premium, 

Divide  the  sum  for  wJiich  the  property  is  to  be  insured 
hy  1 —  the  rate  ^,  written  as  a  decimal. — (Yide  214.) 

EXAMPLES. 

1.  Shipped  a  cargo  of  flour  from  New  York  to  Mata- 
moras,  valued  at  §23940.  For  what  sum  must  it  be  in- 
sured to  cover  the  value  of  the  flour  and  the  premium, 
the  rate  of  insurance  being  5  ^  ?  Ans.  §25200. 

2.  For  what  sum  must  I  insure  §45000  worth  of  cot- 
ton, shipped  from  Yicksburg  to  New  Orleans,  so  as  to 
cover  the  cotton  and  premium,  the  rate  of  insurance 
being  2 J  %  ?  Ans.  §46153  |f 

3.  The  premium  for  insuring  a  house  is  ^50,  including 
§2.00  for  the  policy.  The  rate  of  insurance  is  If  %. 
What  is  the  value  of  the  house  ?— (Yide  212.) 

Ans.  §3000. 


212  PERCENTAGE. 

318.  Commission  is  a  sum  allowed  to  a  Commission 
Merchant,  Agent,  or  Factor,  by  a  Principal,  for  his 
services  in  buying  or  selling  goods.  The  Agent,  if  re- 
siding in  a  different  part  of  the  country,  or  in  a  foreign 
country,  is  called  a  Consignee;  the  goods  shipped,  a 
Consignment;  the  Principal,  a  Consignor. 

The  commission  is  usually  a  given  percentage  of  the 
money  involved  in  the  purchase  or  sale. 

210.  To  find  the  commission  on  a  given  sum,  at  a 
given  rate  per  cent.. 

Multiply  the  given  sum  hy  the  given  rate  ^,  written  as 
a  decimal. — (Vide  210.) 

EXAMPLES. 

1.  I  receive  an  order  in  Mobile,  from  Liverpool,  for 
the  purchase  of  300  bales  of  cotton.  The  cotton  I  buy 
weighs  on  an  average  500  pounds  to  the  bale,  and  costs 
in  Mobile  11 J  cts.  per  lb.  What  is  my  commission,  at 
the  rate  of  1^%  on  the  cost  of  the  cotton? 

Ans.  §253.12^. 

2.  A  commission  merchant  sells  goods  to  the  amount 
of  §4375,  on  which  he  receives  a  rate  of  2  ^.  'To  what 
does  his  commission  amount?  Ans.  |87.50. 

3.  What  is  the  commission  on  a  purchase  of  50  bales 
of  cotton,  at  500  pounds  to  the  bale,  and  costing  10|  cts. 
per  pound,  the  commission  being  IJ  %  ? 

Ans.  §45.39Jg. 

330.  To  find  the  commission  when  the  amount  in- 
cludes the  sum  to  be  invested,  and  also  the  commission, 

Divide  the  given  amount  by  l-\-  the  rate  ^,  ivritten  as 
a  dscimal,  and  subtract  the  result  from  the  given  amount. 


<^  ■^^ 


EXAMPLES. 


•L3 


1.  I  receive  in  Mobile  |7500,  with  which  to  purchase 
cotton.  My  commission  is  to  be  2  ^  on  the  purchase, 
which  is  to  be  deducted  from  the  money.  What  is  my 
commission?  Ans.  $147.05] |. 

2.  Suppose  I  had  received  $22400,  with  the  under- 
standing that  my  commission  should  be  2j  ^  on  the 
amount  purchased.  AVhat  amount  should  I  have  to  ex- 
pend for  cotton?  Ans.  $21853.658. 

3.  I  send  a  commission  merchant  $1000,  with  which 
to  buy  cotton,  after  deducting  his  commission  of  5  J^  on 
the  money  invested.  What  amount  is  invested  in  cotton, 
and  how  much  is  retained  as  commission? 

Ans.  $952.38  and  $47.62. 

221.  Stock  is  money  belonging  to  a  collection  of  in- 
dividuals, called  a  Corporation,  authorized  by  law  to  do 
business  together. 

(1.)  The  owners  of  the  stock  are  called  Stockholders. 

(2.)  A  Share  is  one  of  the  equal  parts  into  which  the 
stock  is  divided.     Such  a  part  is  usually  $100. 

(3.)  A  Certificate  is  a  written  evidence  of  ownership 
of  stock. 

(4.)  The  par  value  of  stock  is  the  number  of  dollars 
mentioned  in  each  share. 

(5.)  The  market  value  of  stock  is  the  number  of  dol- 
lars a  share  w^ll  bring  when  sold  for  cash. 

(6.)  Stock  is  above  par,  or  below  par,  according  as  the 
market  value  is  above  or  below  the  par  value. 

(7.)  If  above  par,  stock  is  at  a  premium;  if  below 
par,  it  is  at  a  discount. 


sft 


PERCENTAGE. 


22S.    To  find  the  value  of  stock  at  a  given  rate  per 
cent,  premium, 

Multiply  the  par  value  hy  l-\-  the  rate  ^    ivritten  as  a 
decimal. 

EXAMPLES. 

1.  What  is  the  value  of  17  shares  of  stock,  at  5  ^     , 

? 


premium 


OPERATION. 

11700 
1.05 


85.00      ^"^^  ■/ 
1700        ^-^    '^^ 


?^ 


$1785.00    Ans.     (Vide  210,  Ex.  17.) 

2.  What  is  the  value  of  14  shares  of  railroad  stock, 
at  a  premium  of  7  %  ?  Ans.  $1498. 

3.  What  is  the  value  of  $32000  in  State  bonds,  at  a 
premium  of  J  ^  ?  '  Ans.  $32040. 

223.  To  find  the  value  of  stock  at  a  given  rate  per 
cent,  discount. 

Multiply  the  par  value  hy  1 —  the  rate  %  written  as  a 
decimal. 

EXAMPLES. 

1.  What  is  the  value  of  17  shares  of  stock  at  5  % 
discount  ? 

OPERATION. 
$1700 

.95 


85.00 
1530.0 

$1615.00 


PERCENTAGE.  215 

2.  What  is  the  value  of  14  shares  of  railroad  stock,  at 
a  discount  of  7  %  ?  Ans.  §1302. 

3.  What  is  the  value  of  12  J  shares  of  stock,  at  a  dis- 
count of  14  %  ?— (210,  Ex.  12.)  Ans.  |1075. 

4.  What  is  the  value  of  |13000  in  State  bonds,  at  a 
discount  of  8%  ?  Ans.  $11960. 

5.  I  invest  $10000  in  railroad  stock  at  the  par  value. 
In  a  year  the  stock  depreciates  3  fo,  and,  fearing  a 
further  decline,  I  sell  all  my  certificates.  What  is  my 
loss?  Ans.  $300. 

6.  If  I  buy  15  shares  of  stock  at  a  premium  oi  Sfo, 
and  sell  at  a  discount  of  3%,  what  do  I  lose? 

Ans.  $90. 

7.  If  I  buy  18  shares  of  railroad  stock  at  5  ^  below 
par,  and  sell  at  7  %  above  par,  what  do  I  gain  ? 

!S34.  Brokerage  is  the  percentage  charged  by  money 
dealers,  called  Brokers,  for  negotiating  Bills  of  Exchange. 
Brokers  all  deal  in  stocks  and  other  monetary  matters. 

225.    To  find  the  brokerage  on  a  given  sum. 
Multiply  the  sum  hy  the  rate    fo    written  as  a  decimal. 

EXAMPLES. 

1.  Wishing  to  rais,e  an  amount  of  money,  I  sell  to  a 
broker  100  shares  of  railroad  stock  at  a  discount  of  i  %. 
What  is  the  amount  of  brokerage?  Ans.  $25.00. 

2.  What  must  I  pay  a  New  Orleans  broker  for  cashing 
bills  on  New  York  to  the  amount  of  $5000,  brokerage 
at  the  rate  of  >  %  ?  Ans.  $25.00. 

22%.  Profit  and  Loss  are  terms  used  by  merchants 
and  other  business  men  in  reference  to  the  purchase  and 
sale  of  goods. 


2r6  PERCENTAGE. 

(1.)  The  cost  is  the  price  paid  for  an  article. 

(2.)  The  selling  price  is  the  amount  received  for  an 
article. 

(3.)  The  profit  is  the  amount  received  less  the  cost. 

(4.)  The  loss  is  the  cost  less  the  amount  received. 

2217.  To  find  the  profit  or  loss  when  the  cost  price  and 
the  rate  per  cent,  of  profit  or  loss  are  given, 

Multiply  the  cost  price  by  the  rate  fo  written  as  a 
decimal.     The  result  will  be  the  profit  or  loss. 

EXAMPLES. 

1.  A  merchant  bought  goods  for  ^500,  and  sold  them 
at  a  profit  of  12^.     What  does  he  clear?    Ans.  $60. 

2.  If  I  buy  goods  for  |750,  and  sell  them  at  a  profit 
of  33A  %,  what  do  I  clear  ?  Ans.  ,^250. 

3.  What  is  the  profit  on  oil,  valued  at  $175,  retailed 
at  25%  above  the  cost?  Aiis.  $43.75. 

4.  Buy  sugar  for  $700,  $800,  and  $1000;  clear  25% 
on  the  first  lot,  33^  %  on  the  second,  and  lose  50  %  on 
the  third.     How  did  I  come  out  of  the  trade? 

Ans.  Lost  $58 J. 

228.  To  find  the  rate  per  cent,  of  profit  or  loss  when 
the  cost  and  selling  prices  are  givQn, 

Divide  the  difference  hetiveen  the  cost  and  selling  prices 
hy  the  cost  price.     Change  the  quotient  by  209. 

EXAMPLES. 

1.  If  I  buy  goods  for  $500,  and  sell  the  same  for 
$560,  what  is  the  rate  %  of  profit? 

2.  If  I  buy  a  quantity  of  flour  for  $750,  and  sell  it 
for  $1000,  what  is  the  rate  %  of  profit? 


PERCENTAGE.  £17 

f 

3.  If  I  buy  flour  at  |4  per  barrel,  and  sell  it  at  §5.50 
per  barrel,  what  is  the  rat^  fo  of  profit? 

OPERAftoNS. 

(1.)  (^.)  (3.) 

§560  11000  $5.50 

500  750  4.00 


500)  60.00(.12         750)  250.00(.33J  4.)1.50(.37i 

60.00  250.00  1.50 

Arts.  12  fc-  Ans.  33|%.  Ans.S1l%> 

4.  If  I  purchase  tea  at  60  cts.  per  pound,  and  sell  it 
at  90  cts.,  what  is  the  rate  ^  of  profit?      Ajis.  50^. 

5.  If  I  buy  40  yards  of  broadcloth  at  §2.50  per  yard, 
and  sell  the  whole  for  $120,  what  is  the  rate  ^  of  profit? 

Ans.  20%. 

6.  I  have  in  my  storehouse  300  barrels  of  damaged 
flour,  which  cost  me  §1450.  I  am  willing  to  sell  the  lot 
at  §4  per  barrel.     What  w^ould  be  the  rate  fo  of  loss  ? 

Ans,  nj^fo. 

7.  Cost  price  §1.20,  selling  price  §1.50.  Rate  %  of 
profit?  Ans,  229,  Ex.  5. 

8.  Cost  price  §1.25,  selling  price  §1.75.  Bate  %  of 
profit  ? 

9.  Cost  price  §1.40,  selling,  price  §2.00.  Rate  %  of 
profit  ? 

10.  Cost  price  §4.50,  selling  price  §6.00  Rate  %  of 
profit  ? 

11.  Cost  price  §6.00,  selling  price  §4.50.  Rate.^  of 
loss? 

12.  Cost  price  §2.00,  selling  price  §1.40.  Rate  %  of 
loss? 

19 


PERCENTAGE. 

229.  To  find  the  selling  price,  when  the  cost  price  is 
known,  so  that  a  given  per  cent,  may  be  made  or  lost, 

(1.)  If  profit  is  to  be  made,  multiply  tlie  cost^rice  hy 
l-\-  the  rate  ^,  ivritten  as  u  decimal.     (Vide  222.)    . 

(2.)  If  loss  is  to  be  susfained,  multiply  the  cost  price 
hy  1 —  the  rate  fo,  ivritten  as  a  decimal.     (Vide  223.) 

EXAMPLES. 

1.  I  buy  goods  for  ^500,  and  propose  to  clear  12  fo- 
What  must  be  the  selling  price?     |500xl.l2. 

Ans.  $560. 

2.  If  I  buy  flour  for  §750,  and  in  the  sale  of  it  clear 
33|  ^,  what  is  my  selling  price? 

$750X1.33J=§750X-|.  Ans.  |1000. 

3.  I  buy  cloth  at  §4  per  yard,  and  wish  to  make  a 
profit  of  37|  %  on  the  cost.  What  must  be  my  selling 
price  per  yard?  §4Xl.37i-=PX  V- 

Ans.  $5.50 

4.  I  have  in  store  300  barrels  of  flour,  which  cost 
$1450.  It  being  damaged,  I  am  willing  to  lose  VI  J^fc 
What  must  I  charge  per  barrel?  Ans.  $4. 

5.  Cost  price  $1.20,  rate  %  of  profit  25.  What  is 
the  selling  price  ? 

6.  Cost  price  $1.25,  rate  %  of  profit  40.  What  is 
the  selling  price  ? 

7.  Cost  price  $1.40,  rate  %  of  profit,  42f .  What  is 
the  selling  price? 

8.  Cost  price  $4.50,  rate  %  of  profit  33 J.  What  is 
the  selling  price? 

9.  Cost  price  $6.00,  rate  %  of  loss  25.  What  is  the 
selling  price? 


PERCENTAGE.  219 

10.  Cost  price  12.00,  rate  %  of  loss  30.  What  is  the 
selling  price? 

11.  To  make  12 »  %  profit,  for  how  much  must  I  sell 
cloth  that  cost  16  cts.  per  yard?  24  cts.?  32  cts.?  64  cts.? 
72  cts.?  88  cts.?  Am.  %  of  16  cts.=18  cts.,  etc. 

12.  To  make  16f  %  profit,  what  must  a  merchant 
mark  calico  that  cost  36  cts.  per  yard?  42  cts.?  54  cts.? 
72  cts.?  11.26?  $1.50?  $1.80?     . 

13.  To  make  33^^,  what  must  I  mark  books  which 
cost  24  cts.?  27  cts.?  30  cts.?  42  cts.?  §1.02?  $1.05? 
$1.08?  $1.24?  $1.44? 

230.  To  find  the  cost  price,  when  the  selling  price 
and  the   rate  per  cent,  of  profit  or  loss  are  given. 

(1.)  If  a  profit  has  been  made,  divide  the  Belling  'price 
hy\-\-  the  rate  %   written  as  a  deciftiaL     (Vide  213.) 

(2.)  If  a  loss  has  been  sustained,  divide  the  selling 
pnce  hy  1 —  the  rate  %  written  as  a  decimal,  (Vide  214.) 

EXAMPLES. 

1.  Selling  price  $560,  rate  %  of  profit  12.  What  is 
the  cost  price? 

2.  Selling  price  $1000,  rate  %  of  profit  33^.  What 
is  the  cost? 

3.  Selling  price  $5.50,  rate  ^  of  profit  37|.  What 
is  the  cost? 

4.  Selling  price  $1.50,  rate  %  of  profit  25.  What 
is  the  cost? 

5.  Selling  price  $1.75,  rate  %  of  profit  40.  What 
is  the  cost? 

6.  Selling  price  $2.00,  rate  %  of  profit  42f .  What 
is  the  cost? 


220  PERCENTAGE. 

7.  Selling  price  $4:.50,  rate  %  of  loss  25.  What  is 
the  cost? 

231.  Duties  or  Customs  are  sums  of  money  assessed 
by  government  upon  imported  goods. 

(1.)  Specific  duties  are  assessed  upon  goods  at  a  cer- 
tain rate  per  hogshead,  gallon,  bale,  etc.,  with  no  refer- 
ence to  their  value. 

(2.)  Ad  valorem  duties  are  a  certain  percentage  of  the 
cost  of  goods. 

(3.)  An  Invoice  is  a  written  account  of  the  goods  of  a 
cargo  containing  a  statement  of  the  cost  of  each  article 
in  the  currency  of  the  country  whence  imported. 

EXAMPLES. 

1.  The  invoice  of 'a  cargo  of  goods  which  arrived  in 
Mobile  from  Liverpool,  contained  the  following  among 
other  items : 

325  yd.  Broadcloth  cost  26  s.         sterling  per  yd. 
623  yd.  Muslin  "       4  s.  "  " 

600  yd.  Lace  "       Is.  lOd.     "  " 

975  yd.  Carpeting      "       6  s.  "  " 

1280  yd.  "  "       4s.  8d.       "  " 

The  duty  on  the  broadcloth  was  15  ^  ;  on  the  muslin 
12|  fo  ;  on  lace  12 J  ^  ;  carpeting  15  %.  What  was 
the  amount  of  duty  in  United  States  money? 

Ans.  1844.58. 

232.    Miscellaneous  Examples. 
1.  A  and  B  invest  $550  in  a  speculation  of  which  A  fur- 
nishes $330,  and  B  the  balance.     They  gain  $70.     What 
is  the  rate  %  of  profit  on  the  money  invested  ?     What  is 
the  share  of  each  ?        Ans.  12  f\  %-,  A  $42;  B  $28. 


PERCENTAGE.  221 

2.  A  man  failing  owes  A  §175;  B  $500;  C  $600;  D 
1210;  E  $42.50;  F  $20;  and  G  $10.  His  property  is 
sold  for  $934.50.  What  is  the  rate  %  of  loss?  What 
is  the  share  of  each  creditor? 

Ans.  Loss  40  per  cent.     A  loses  $70;  B  $200;  C  $240;  D  $84.00; 
E  $17;  F  $8;  G  $4. 

3.  A  bankrupt  owes  A  $500;  B  $1200;  and  C  S4300. 
The  net  cash  proceeds  of  his  estate  amount  to  only 
$1500.  What  rate  ^  does  he  pay  on  his  debts?  What 
does  each  creditor  receive? 

Am.  2h%,     A  $125;  B  $300;  C  $1075. 

4.  If  the  money  and  effects  of  a  bankrupt  amount  to 
$3361.74,  and  he  is  indebted  to  A  $1782.24,  to  B 
$1540.76,  and  to  C  $2371.17,  how  much  will  each  re- 
ceive?     Ans.  A  $1052.20;  B  $909.64;  C  $1399.90. 

5.  I  send  a  commission  merchant  $1000,  with  which 
to  buy  cotton.  If  I  allow  him  5  %  commission  on  the 
money  sent,  how  much  will  he  have  to  expend  in  cot- 
ton? ^Tis.  $950. 

6.  How  much  railroad  stock  can  be  obtained  for 
$3860,  when  the  stock  is  at  a  discount  of  ^l%t 

Ans.  $4000. 

7.  What  rate  %  of  $700000  is  $700?     Ans.  J^%.  . 

8.  What  rate  %  of  $450000  is  $2250?    Ans.  \%. 

9.  A  certain  town,  whose  property  is  valued  at  $750- 
000,  proposes  to  raise  a  tax  of  $1875.  What  will  be 
the  rate  %  ?  Ans.  J  % ,  or  2^  mills  on  the  dollar. 

10.  AVhat  will  be  the  tax  of  a  man  whose  property 
is  valued  at  $12000,  at  the  rate  of  i  %  ?       Ans.  $30. 

11.  A  city  agrees  to  loan  a  railroad  company  $1000- 
000,  which  amount  is  to  be  raised  on  a  property  valued 


222  PERCENTAGE. 

at  $175000000.     What  is  the  rate  %  of  taxation,  and 
what  does  A  pay,  whose  property  is  taxed  for  |35000  ? 

Ans.  4%.     A  pays  $200. 

12.  Sold  tea  at  90  cts.  per  pound,  and  gained  20  %. 
What  %  should  I  have  gained  had  I  sold  it  for  $1.00 
per  pound?  Am.  33i  %, 

13.  I  sell  tea  at  $1.28  per  pound,  and  thereby  lose 
20  % .  What  would  be  gained  or  lost  ^  by  selling  the 
same  tea  at  $1.68  per  pound?         Am.  Profit  of  5%. 

14.  iBy  selling  coffee  at  67  J  cts.  per  pound  I  make  a 
profit  of  12J%,  but  I  desire  to  make  30%.  What  must 
be  my  selling  price  per  pound?  An%.  78  cts. 

15.  I  bought  a  quantity  of  broadcloth  for  $2.59  per 
yard,  but  on  measuring  it  I  find  it  falls  short  12|%  in 
length.  What  must  be  my  selling  price  per  yard  in 
order  to  clear  12 1%  on  the  real  cpst?        Am.  $3.33. 

16.  I  bought  a  quantity  of  calico  at  40 J  cts.  per  yd.; 
but  10%  of  the  calico  proved  to  be  damaged,  and  10% 
of  the  balance  was  lost  by  bad  debts,  and  yet  I  cleared 
10  %  on  the  cost.  What  was  the  selling  price  per 
yard?  An%.  55  cts. 

INTEREST. 

1333.   Interest  is  a  percentage  paid  for  the  use  of 

money. 

(1.)  The  Principal  is  the  money  for  which  interest  is 
paid. 

(2.)  The  Amount  is  the  sum  of  the  principal  and 
interest. 

(3.)  The  Rate  per  cent,  per  annum  is  the  number  of 
cents  paid  fur  the  use  of  1  tlollar  for  a  year. 


PERCENTAGE.  223 

(4.)  The  Time  is  the  period  for  which  interest  is  paid. 
Thus, 

July  4,  1865,  A  borrowed  of  B  $7250,  agreeing  to 
pay  at  the  rate  of  8^  per  annum.  August  1,  1866, 
he  paid  $623.50. 

The  principal  is  $7250;  the  interest  is  $623.50;  the 
amount  is  $7873.50;  the  rate  per  cent,  per  annum  is 
8;  the  time,  1  yr.  27  da.  (Vide  198,  Ex.  2;  and  202, 
Ex.  55.) 

Resiark  1. — The  rate  per  cent,  in  tlie  various  states  is  estab- 
lished by  law,  and  is  thence  called  the  legal  rate;  a  higher  than 
legal  rate  is  usury.  The  legal  rate  in  Maine,  New  Hampshire, 
Vermont,  Massachusetts,  Rhode  Island,  Connecticut,  New  Jersey, 
Pennsylvania,  Delaware,  Maryland,  Virginia,  North  Carolina, 
Tennessee,  Kentucky,  •••Ohio,  ^Indiana,  •••Illinois,  *Iowa,  "^Nebraska, 
^Missouri,  *Kansas,  ^Arkansas,  -^Mississippi,  Florida,  District  of 
Columbia,  and  debts  in  favor  of  the  United  States,  is  6  per  cent.; 
^Michigan,  New  York,  Minnesota,  Georgia,  and  South  Carolina,  7 
per  cent.;  Alabama  and  Texas,  8  per  cent.;  California,  10  per  cent.; 
Louisiana,  5  per  cent.  By  special  contract,  parties,  in  the  states 
marked  *  can  take  interest  as  high  as  10  per  cent. 

Remark  2. — The  month,  in  computing  interest,  is  regarded  as 
having  30  days.     Custom  has  made  it  lawful.     (Vide  185,  Rem.  6.) 

!334.  To  find  the  interest  of  any  principal  for  1  year, 
at  any  given  rate  per  cent,  per  annum. 

Multiply  the  principal  hy  the  rate  fo  per  annum^  writ- 
ten as  a  decimal. 

EXAMPLES. 

1.  What  is  the  interest  of  $6500  for  1  year,  at  1  % 
per  annum?  2%?  3%?  4%?  etc.,  to  12%? 

Ans.  $65;  $130;  $195;  $260,  etc.  .  .  $780. 

2.  At  12  %  per  annum,  what  is  the  interest  of  $1041| 
forlyear?  $1458i?  $3333i?  $4375?  $6250?  $8333i? 
$20833^?  $6875?         ^ws.  $125;  $175,  etc.  .  $825. 


224  PERCENTAGE. 

3.  At  Sfo  per  annum,  what  is  the  interest  of  §5000 
for  1  year?  $1850?  |9215?  $8000?  $7250?  $5280? 

Ans.  202,  Ex.  51.  .  .  56. 

4.  What  is  the  amount  of  $500,  at  2^  per  annum, 
for  1  year?  $250  at  5%?  $375  at  6%?  $475  at  6%? 
$450  at  7  %  ?  $1250  at  8  %  ?     (Vide  233,  (2.) 

Ans.  $^10;  $262.50.  .  .  $1350. 

5.  What  is  the  amount  of  $1.00,  at  5^  per  annum, 
fori  year?  6%?  7%?  8%?  10%? 

Ans.  $1.05;  $1.06;  $1.07;  $1.08;  $1.10. 

6.  What  is  the  amount  of  $1.05,  at  5%  per  annum, 
for  1  year?  $1.06  at  6%  ?  $1.07  at  7%  ?  $1.08  at  8%  ? 
$1.10  at  10%? 

Ans.  $1.10i;  $1.12/^;  $1.1449;  $1.1664;  $1.21. 
'  235.    To  find  the  interest  of  $1.00,  at  12  %   per 
annum,  for  any  given  time. 

EXAMPLES. 

1.  Find  the  interest  of  $1.00,  at  12%  per  annum,  for 
3  yr.  5  mo.  15  da. 

ANALYSIS. 

(1.)  12  per  cent,  per  annum,  means  12  cents  for  the  use  of  $1 
one  year.     (Vide  233,  (3.) 

(2.)  12  cents  per  year  gives  a  rate  of  1  cent  per  month. 

(3.)  10  mills  (1  cent)  for  30  days  (1  month)  gives  a  rate  of  1 
mill  for  3  days. 

The  interest  of  $1.00  for  3  yr.  5  mo.  is  therefore   3Xl2-f-5      $0.41 
The  interest  of  $1.00  for  15  da.  is         .         .         15-5-3    .     .    0.005 

The  interest  of  $1.00  for  3  yr.  5  mo.  15  da.  is  (202,  Ex.  57.)    $0,415 

Hence, 

(1.)   Call  the  months  in  the  given  time  so  many  cents. 

(2.)  Call  one  third  of  the  days  so  many  mills. 


PERCENTAGE. 


225 


Tlie  sum  of  tliese  results  will  be  the  interest  required. 

Find  the  interest  of  |1.00,  at  12  ^  per  annum,  for 
the  following  times.  The  work  should  be  done  mentally. 
(Vide,  also,  205,  Ex.  56.) 


2.  1  yr.  2  mo.  6  da. 

Ans.  §0.142. 

3.  1  yr.  3  mo.  9  da. 

Ans.  $0,153. 

4.  2  yr.  4  mo.  12  da. 

.4ns.  $0,284. 

5.  3  yr.  6  mo.  15  da. 

Ans.  $0,425. 

6.  4  yr.  1  mo.  18  da. 

Ans.  $0,496. 

7.  5  yr.  3  mo.  21  da. 

Ans.  $0,637. 

8.  6  yr.  7  mo.  24  da. 

Ans.  $0,798. 

9.  1  yr.  3  mo.  29  da. 

Ans.  $0.159f. 

10.  Oyr.  4  mo.  12  da. 

Ans.  $0,044. 

11.  Oyr.  8  mo.  3  da. 

Ans.  $0,081. 

12.  0  yr.  0  mo.  24  da. 

Ans.  $0,008. 

13.  20  yr.  1  mo.  3  da. 

Ans.  $2,411. 


14.  1  yr.  1  mo.  27  da. 

A71S.  $0,139. 

15.  8yr.  4  mo.  0  da. 

Ans.  $1.00. 

16.  2  yr.  1  mo.  1  da. 

Ans.  $0.25O|. 

17.  3  yr.  2  mo.  2  da. 

Ans.  $0.380f. 

18.  4  yr.  3  mo.  4  da. 

Ans.  $0.511|. 

19.  5  yr.  7  mo.  11  da. 

A71S.  $0.673f. 

20.  3  yr.  4  mo.  25  da. 

A71S.  $0.4081-. 

21.  5  yr.  3  mo.  9  da. 

.4ns.  $0,633. 

22.  0  yr.  1  mo.  1  da. 

A71S.  $0.0101 

23.  0  yr.  8  mo.  2  da. 

Ans.  $0.080f. 

24.  12  yr.  0  mo.  0  da. 

Ans.  $1.44. 

25.  25  yr.  5  mo.  0  da. 


Ans.  $3,052. 

S30.  To  find  the  interest  on  any  principal,  for  any 
time,  at  any  rate  per  cent,  per  annum, 

Multiply  the  principal  by  the  interest^of  §1  at  12%  per 
annum  for  the  given  time.     (Vide  235.) 

(1.)   The  interest  at  1%  is  one  twelfth  that  at  12%. 

(2.)  The  interests  at  2%,  3%,  4%,  and  6%  are,  re- 
spectively, one  sixth,  one  fourth,  one  third,  and  one  half 
that  at  12%. 


226  '  PERCENTAGE. 

(3.)  The  interests  at  S%  and  9  %  are,  respectively,  two 
thirds  and  three  fourths  that  at  12  %. 

(4.)  The  interest  at  any  rate  whatever  may  he  found  hy 
multiplying  that  at  Ifo  hy  the  numher  representing  the 
rate  %  per  annum. 

EXAMPLES. 

1.  What  is  the  interest  of  $421.40  for  3  yr.  5  mo. 
15  da.,  at  the  rate  of  6^  per  annum? 

OPERATION. 

1421.40 
(Vide  235,  Ex.  1.)  .415 

210700 
42140 

168560 


(Vide  235,  (2.)  2)174.88100 

(Vide  202,  Ex.  59.)  |87.4405     Ans. 

2.  What  is  the  interest  of  $19.35  for  lyr.  2  mo.  6  da., 
at  the  rate  of  6^  per  annum?  Ans.  $1,374. 

3.  What  is  the  interest  of  $17.21  for  1  yr.  3  mo.  9  da., 
at  the  rate  of  6^  per  annum?  Ans.  $1.316565. 

4.  What  is  the  interest  of  $140.10  for  2  yr.  4  mo. 
12  da.,  at  the  rate  of  6  ^  per  annum  ? 

Ans.  $19.8942. 

5.  What  is  the  interest  of  $75.15  for  3  yr.  6  mo. 
15  da.,  at  the  rate  of  6^  per  annum? 

Ans.  $15.96||. 

6.  What  is  the  interest  of  $1000  for  4  yr.  1  mo. 
18  da.,  at  the  rate  of  6%  per  annum? 

Ans.  $248. 


PERCENTAGE.  227 

7.  What  is  the  interest  of  |175  for  5  jr.  3  mo.  21  da., 
at  the  rate  of  6%  per  annum?  Ans.  |55.73|. 

8.  What  is  the  interest  of  $2141  for  6  yr.  7  mo.  24 
da.,  at  the  rate  of  6%  per  annum?       Aiis.  §854.259. 

9.  What  is  the  interest  of  $1041|  for  1  yr.  3  mo.  15 
da.,  at  the  rate  of  6^  per  annum?      Ans.  |80.72iJ. 

10.  What  is  the  interest  of  |1458|  for  1  yr.  4  mo.  21 
da.,  at  the  rate  of  6^  per  annum?     Ans.  $121.77y'2. 

11.  What  is  the  interest  of  ?3333|  for  1  yr.  5  mo. 
7  da.,  at  the  rate  of  6  ^  per  annum  ? 

Ans.  $287.22f . 

12.  What  is  the  interest  of  $4375  for  2  yr.  3  mo.  12 
da.,  at  the  rate  of  6%  per  annum?      Ans.  $599. 37^. 

13.  What  is  the  interest  of  $421.40,  from  January  1, 

1865,  to  June  16,  1868,  at  6%  per  annum?  3^?  4%? 

Last  Ans.  $58.293|.. 

14.  What  is  the  interest  of  $19.35,  from  April  4, 

1866,  to  June  10,  1867,  at  2  %  ?  3  %  ?  4  %  ? 

Last  Ans.  $0,916. 

15.  What  is  the  interest  of  $25.14,  from  February  6, 
1866,  to  April  3,  1867,  at  6%  ?  3%  ?  4^  ? 

First  Ans.  $1,747. 

16.  What  is  the  interest  of  $200,  from  March  14, 
1865,  to  July  14, 1873,  at  6  %  ?  3  %  ?  4  %  ? 

First  Ans.  $100.00.     • 

17.  What  is  the  interest  of  $525,  from  May  17, 1868, 
to  June  18,  1870,  at  6%?  at  3%?  at  4%? 

First  Ans.  $65.71i. 

18.  What  is  the  interest  of  $700,  from  August  14, 
1865,  to  October  16,  1868,  at  8%?  at  2%?  at  4%? 

•     Ans.  $177,644 


228  PEIICENTAGE. 

19.  What  is  the  interest  of  §925.16,  from  October  15, 

1865,  to  January  19,  1870,  at  8%?  at  2%?  at  4%? 

Ans.  $315,377. 

20.  What  is  the  interest  of  $375,  from  December  8, 

1869,  to  Julj  19,  1875,  at '8%?  at  2%?  at  4^? 

Ans.  $168,416. 

21.  What  is  the  interest  of  $1400,  from  November  4, 
1868,  to  March  29,  1872,  at.8%?  at  2%?  at  4%? 

Ans.  $381,111. 

22.  What  is  the  interest  of  $2000,  from  September 
10,  1869,  to  February  5,  1873,  at  8%?   at  2%?  at  4%? 

Ans.  $544,444. 

23.  What  is  the  interest  of  $4062.50,  from  July  5, 

1866,  to  October  15,  1868,  at  8^  per  annum? 

Ans.  202,  Ex.  39. 

24.  What  is  the  interest  of  $1562.50,  from  June  6, 

1870,  to  September  21,  1871,  at  8%?  2%?  4%? 

Ans.  202,  Ex.  40. 

25.  What  is  the  interest  of  $2187.50,  from  August 
10,  1868,  to  January  1,  1870,  at  8  %  ?  2  %  ?  4  %  ? 

Ans.  202,  Ex.  41. 

26.  What  is  the  interest  of  $8000,  from  January  1, 
1866,  to  June  8,  1867,  at  5  %  ?  10  %  ? 

Ans.  202,  Ex.  42. 

27.  What  is  the  interest  of  $10500,  from  February 
1,  1867,  to  May  13,  1869,  at  5  %  ?  10  %  ? 

Ans.  202,  Ex.  43. 

28.  What  is  the  interest  of  $15000,  from  March  4, 
1868,  to  October  23, 1871,  at  5  %  ?  10  %  ? 

Ans.  202,  Ex.  46. 


PERCENTAGE.  229 

29.  What  is  the  interest  of  ^20000,  from  April  9, 
1867,  to  February  6,  1873,  at  5%?  at  10%? 

Ans.  202,  Ex.  47. 

30.  What  is  the  interest  of  |50000,  from  May  16, 

1865,  to  May  11,  1870,  at  5%?  at  10%? 

Am.  $12465.27?. 

31.  What  is  the  interest  of  $425.30,  from  March  4, 

1866,  to  May  19,  1868,  at  7%?  at  3j%? 

Ans.  $65,744. 

32.  What  is  the  interest  of  $510.83,  from  March  21, 

1867,  to  December  30,  1867,  at  7%?  at  3^-%? 

Ans.  $27,713. 

33.  What  is  the  interest  of  $170,  from  June  19, 1865, 
to  July  1,  1866,  at  7%?  at  3^%?  Ans.  $12,296. 

34.  What  is  the  interest  of  $966,  from  January  1, 

1867,  to  March  20,  1869,  at  7%?  at  3j%? 

Ans.  $150,078. 

35.  What  is  the  interest  of  $213.27,  from  August  15, 
1872,  to  March  13,  1875,  at  7%?  at  3J%? 

Ans.  $38,483. 

36.  What  is  the  interest  of  $426.50,  from  Sept.  4, 

1868,  to  May  4, 1874,  at  9  %?  4i-  %?    Ans.  $217,515. 

37.  What  is  the  amount  of  $164.06,  from  July  4, 
1864,  to  February  15,  1870,  at  3%?  1^%?  6%?  9%? 
4J  %?     (Vide  233,  (2.)      Ans.  $191.69 ;  $177.87,  etc. 

38.  What  is  the  amount  of  $120.10,  for  8  yr.  4  mo., 
at  12%  per  annum?  100  yr.,  at  1%?  50  yr.,  at  2%? 

Ans.  $240.20. 

39.  What  is  the  amount  of  $120.10,  for  12  yr.  6  mo., 
at  8  %  per  annum?  25  yr.,  at  4  %  ?  33  yr.  4  mo.,  at 
3  %  ?  Ans.  $240.20. 


230  PERCENTAGE. 

40.  What  is  the  amount  of  $120.10,  for  16  yr.  8  mo., 
at  6%  per  annum?  for  20  yr.,  at  5%?    12  yr.,  at  8|%? 

Ans.  $240.20. 

41.  What  is   the   amount  of  $450,  from  January  1, 
1865,  to  March  16,  1865,  at  8^  per  annum? 

Ans.  $457.50. 

42.  What  is  the  amount  of  $382.50,  from  March  16, 

1865,  to  January  1,  1866,  at  8^  per  annum? 

An§.  $406,725. 

43.  What  is  the  amount  of  $306,725,  from  January 
1,  1866,  to  April  4,  1866,  at  8%  per  annum? 

Ans.  $313,064. 

44.  What  is  the  amount  of  $113,064,  from  April  4, 

1866,  to  January  1,  1867,  at  8^  per  annum? 

-  Ans.  $119,772. 

45.  What  is  the  amount  of  $700,  from  January  1, 

1865,  to  July  28,  1865,  at  6%  per  annum? 

Ans.  $724.15. 

46.  What  is  the  amount  of  $624.1 5,  from  July  28,1865, 
to  April  4,  1866,  at  6^  per  annum?     Ans.  $649.74. 

47.  What  is  the  amount  of  $149.74,  from  April  4, 

1866,  to  January  1,  1867,  at  6^  per  annum? 

Ans.  $156.40. 

48.  What  is  the  amount  of  $3000,  from  Jan.  1,  1865, 
to  April  1,  1-865,  at  10%  per  annum?       Ans.  $3075. 

49.  What  is  the  amount   of   $2075,  from   April  1, 
1865,  to  January  1,  1866,  at  10%  per  annum? 

Ans.  $2230.625. 

50.  What  is  the  amount  of  $1230.625,  from  January 
1,  1866,  to  January  1,  1867,  at  10%  per  annum? 

Ans.  $1353.68f. 


PERCENTAGE.  231 

51.  What  is  tlie  amount  of  $620.53,  for  4  mo.  6  da., 
at  5%  per  annum?  Arts.  |631.39. 

52.  What  is  the  amount  of  |1123.60,  for  8  mo.  15  da., 
at  6%  per  annum?  Ans.  |1171.353. 

53.  What  is  the  amount  of  $1531.301,  for  3  mo. 
24  da.,  at  7%  per  annum?  Ans.  $1565.24. 

54.  What  is  the  amount  of  $4709.25,  for  5  mo.  27 
da.,  at  10%  per  annum?  Ans.  $4940.788. 

2S*7.  To  find  interest  for  days,  counting  365  to  the  year, 
Multiply  one  year's  interest  (vide  234)  by  the  number 
of  days  in  the  time,  and  divide  the  product  by  365. 

EXAMPLES. 

1.  What  is  the  interest  of  $6500,  from  April  15  to 
December  15, 1866,  at  6  %?    (Vide  185,  Rem.  6 ;  Table.) 

$390X244--365.  J^ns.  $260.71. 

2.  At  6fo  per  annum,  what  is  the  interest  of  $375 
from  January  1  to  March  15?  $22.50  x  73 -4-365  =  i  of 
$22.50.  Ans.  $4.50. 

3.  What  is  the  interest  of  $1000,  for  365  days,  at  the 
rate  of  6%  for  360  days?     $60Xftf=$60Xf|. 

Ans".  $60,831. 

4.  What  is  the  interest  of  $1000,  for  360  days,  at  the 
rate  of  6  ^  for  365  days  ?     $60Xf  |f=$60X-ff • 

Ans.  $59.18. 

5.  What  is  the  interest  of  $500000,  for  365  days,  at 
the  rate  of  6%  for  360  days?  Ans.  $30416f. 

6.  What  is  the  interest  of  $500000,  for  360  days,  at 
the  rate  of  6%  for  365  days?  Ans.  $29589^^5. 

7.  What  is  the  interest  of  $500000,  for  90  days,  at 
the  rate  of  6%  per  annum  of  360  days?    Ans.  $7500. 


232  PERCENTAGE." 

8.  What  is  the  interest  of  $500000,  for  90  days,  at 
the  rate  of  6%  per  annum  of  365  days? 

Ans.  $7397.26. 

9.  What  is  the  interest  of  a  §1000  bond,  for  75  days, 
at  the  rate  of  7.30  fo  per  annum  of  365  days  ?     20  cts. 

per  day.  Ans.  $15.00. 

10.  What  is  the  interest  of  a  $100  bond,  for  67  days, 
at  the  rate  of  7.30  fo  per  annum  of  365  days  ?  2  cts. 
per  day.  Ans.  $1.34. 

11.  What  is  the  interest  of  a  $10000  bond,  for  93 
days,  at  the  rate  of  7.30  %  per  annum  of  365  days? 

$2  per  day.  AnS.  $186. 

Remark. — In  New  York,  interest  for  years  and  months  is  com- 
puted by  the  rule  under  sec.  236,  but  for  the  odd  days  by  the  rule 
under  this  section,  237. 

12.  What  is  the  interest  on  a  note  of  $1000,  in  New 
York,  having  run  from  March  4,  1865,  to  March  25, 
1866?  •  ^ns.  $74.03. 

13.  What  wbuld  be  the  interest  of  the  above  note 
computed  in  Kentucky?  Ans.  $63.50. 

PROBLEMS   IN   INTEREST. 

238.  To  find  the  principal,  when  the  time,  rate  per 
cent.,  and  interest  are  given. 

Divide  the  given  interest  hy  the  interest  o/i  $1.00  at 
the  given  rate  and  time. 

EXAMPLES. 

1.  The  interest  of  a  certain  sum,  at  6  %  per  annum,  for 
Syr.  5 mo.  15 da.  is  $87.4405.  What  is  the  principal? 
(Vide  236,  Ex.  1.)     $87.4405-.2076.         Ans.  $421.40. 


PERCENTAGE.  233 

2.  The  interest  of  a  certain  sum  at  6  ^  per  annum, 
for  1  yr.  2 mo.  6 da.  is  ^1.374.     What  is  the  principal? 

Ans.  236,  Ex.  2. 

239.  To  find  the  principal,  when  the  time,  rate  per 
cent.,  and  amount  are  given, 

Divide  the  given  amount  bij  the  amount  of  §1.00  at 
the  given  rate  and  time. 

EXAMPLES. 

1.  The  amount  of  a  certain  sum  at  6%  per  annum, 
for  3yr.  5  mo.  15  da.  is  §508.8405.  What  is  the  prin- 
cipal?     $508.8405--1.2075.  Ans.  $421.40. 

2.  The  amount  of  a  certain  sum,  at  8^  per  annum, 
from  January  1,  1865,  to  March  16,  1865,  is  $457.50. 
What  is  the  principal  ?  Ans.  236,  Ex.  41. 

3.  The  amount  of  a  certain  sum,  at  10%  per  annum, 
from  January  1, 1866,  to  January  1, 1867,  is  $1353. 68|. 
What  is  the  principal?  Ans.  236,  Ex.  50. 

240.  To  find  the  time,  when  the  principal,  interest, 
and  rate  per  cent,  are  given. 

Divide  the  given  interest  by  the  interest  on  the  principal 
at  the  given  rate  for  1  year. 

EXAMPLES. 

1.  The  interest  of  $421.40,  at  6  %  per  annum,  is 
$87.4405.     What  is  the  time?       87.4405 --25.284 =3.458^ 

yr.=3yr.  5mo.  loda.      (Vide  195.) 

2.  The  interest  of  $75.15,  at  6  %  per  annum,  is 
$15.96|-§.     What  is  the  time? 

Ans.  236,  Ex.  5. 
20 


234  PERCENTAGE. 

3.  The  interest  of  §525  to  June  18,  1870,  at  6  % 
per  annum,  is  $65.71 1.  What  is  the  date  from  which 
interest  is  computed?  Ans.  236,  Ex.  17. 

4.  The  interest  on  §100  is  also  §100.  What  has  the 
time  been  at  the  rate  of  1  %  per  annum?  2%?  3%? 
4%?  5%?  6%? 

5.  The  interest  on  §120.10  is  also  §120.10.  What 
has  the  time  been  at  the  rate  of  6  ^  per  annum? 
8%?  12%?  Ans.  236,  Ex.  38,  39,  40. 

6.  In  what  time  will  a^iy  sum  double  itself  at  a  rate  of 
5%  per  annum?  7%?    9%?   4i%? 

Ans.  20  yr.,  etc. 

7.  In  what  time  will  any  sum  triple  itself  at  the  rate 
of  6%  per  annum?  Aiis.  33 yr.  4 mo. 

S41.  To  find  the  rate  per  cent.,  when  the  principal, 
interest,  and  time  are  given, 

Divide  the  given  interest  hy  the  intei^est  on  the  principal 
at  Ifo  p^T  annum  for  the  given  time. 

EXAMPLES. 

1.  The  interest  of  §421.4t)  for  3yr.  5  mo.  15  da.  is 
§87.4405.  What  is  the  rate  %  per  annum?  87.4405-- 
14.5734^  Ans.  6  % . 

2.  The  interest  of  §700  from  August  14,  1865,  to 
October  16,  1868,  is  §177.64|.  What  is  the  rate  % 
per  annum?  Ans.  236,  Ex.  18. 

3.  The  interest  of  §750  for  3  yr.  4  mo.  is  §162.50. 
What  is  the  rate  %  per  annum?  Ans.  6j%. 

4.  The  interest  of  §950  for  2  yr.  4  mo.  20  da.  is 
§238.29 J.  What  is  the  rate  %  per  annum?  (Vide  202, 
Ex.  58.)  '         Ans.  10J%. 


PERCEXTAGE.  235 

PRESENT    Yv^OKTH. 

242.  The  Present  Worth  of  a  sum  of  money  due 
at  some  future  time,  is  a  principal  ^vliicli,  being  put  at 
interest  at  the  time  \)f  payment,  Avill  amount  to  the  sum 
at  the  time  it  is  due. 

The  Discount  is  the  diiference  between  the  present 
worth  and  the  sum  of  money  due. 

243.  To  find  the  present  worth  of  a  sum  of  money 
for  a  given  time  and  rate  per  cent,  per  annum, 

Divide  the  given  sum  by  the  amount  of  fl.OO  at  the 
given  rate  and  time.  (Vide  239.)  The  quotient  is  the 
present  worth. 

EXAMPLES. 

1.  What  is  the  present  worth  of  |508.8405,  due  in  Syr. 
5  mo.  15  da.,  at  the  rate  of  G  ^  per  annum?  §508.8405-^- 
1.2075.  Alls.  S421.40. 

2.  What  is  the  discount  of  §457.50,  due  March  16, 
1865,  but  paid  January  1,  1865,  at  8%  per  annum? 

Ans.  ^7.50. 

3.  What  is  the  discount  on  §1353.68 1,  due  January 
1,  1867,  but  paid  January  1,  1866,  at  10  J/^  per  annum? 

A71S.  $123,061. 

4.  What  is  the  discount  of  $1800,  due  1  yr.  3  mo. 
hence,  at  the  rate  of  6  ^  per  annum  ? 

Ans.  $125,582. 

5.  What  is  the  discount  of  $475,  due  1  yr.  hence,  at 
the  rate  of  7%  per  annum?  Ans.  $31,075. 

6.  What  is  the  discount  of  $2500,  due  3  mo.  hence, 
when  money  is  worth  4J%?  A71S.  $27.81. 


236  PERCENTAGE. 

7.  Four  notes  are  due  as  follows :  §900  in  6  mo. ; 
$2700  in  1  yr.;  |3900  in  1  yr.  6mo.;  and  |4200  in  2yr. 
If  the  notes  are  all  paid  at  the  present  time,  what  will 
be  the  entire  discount,  at  6$^  ?  Ans.  ?951.06. 

8.  What  is  the  discount  of  §400,  due  three  months 
hence,  but  paid  now,  at  the  rate  of  12^  per  annum? 
S%2  r^/ot  6%?  5^,? 

>lns.  §11.65;  §7.84;  §6.88;  §5.91;  §4.94. 

BANK    DISCOUNT. 

244.  Bank  Discount  is  a  deduction  made  by  a  bank 
upon  a  sum  of  money  borrowed  of  it,  at  the  time  the 
money  is  taken. 

(1.)  By  custom,  this  discount  is  the  interest  on  the 
face  of  the  note  for  the  time  it  is  drawn,  increased  by 
three  days,  called  Days  of  Grace, 

(2.)  The  Proceeds  of  a  bank  note  is  what  remains 
after  the  discount  has  been  deducted. 

245.  To  find  the  bank  discount  on  a  note  or  draft, 
Find  the  interest  on  the  face  of  the  note  for  three  days 

more  than  the  time  for  ivhich  it  is  draivn. 

EXAMPLES. 

1.  What  is  the  bank  discount  on  a  note  of  §400,  dis- 
counted for  90  days,  at  8^;  ?  6  %  ?  2  of  $400X.031;  |  of 
$.loox.03i.  Ans.  §8.26f  ;  §6.20. 

2.  Find  the  proceeds  of  a  note  of  §150,  discounted 
forGOdays,  at  6%?   7%?   8%?   10%?     1  of  $i50x.02i; 

-i^jj  of  $150X.021,  etc. 

^715.  §148.42i;  §148.16.1;  §147.90;  §147.37i. 


TEJICENTAGE.  237 

3.  Find  the  proceeds  of  a  note  of  §177.75,  discounted 
for  90  days,  at  6^  per  annum.  Ans.  §175.00. 

4.  Find  the  proceeds  of  a  note  of  §505.305,  discounted 
for  60  days,  at  6^  per  annum.  A7is.  §500. 

5.  Find  the  proceeds  of  a  note  of  §375.00,  discounted 
for  30  days,  at  6^  per  annum.  Ans.  §372. 93| 

246.  To  make  a  note  of  which,  when  discounted,  the 
proceeds  shall  be  a  given  sum. 

Divide  the  given  sum  by  the  proceeds  on  §1.00  at  the 
given  time  and  rate  per  cent,  per  annum. 

EXAMPLES. 

1.  I  wish  to  obtain  §175  for  90  days.     For  what  sum 
must  my  note  be  drawn,  at  6  ^^  per  annum  ? 
1.00-§0.0155=§0.9845,  then  §175-^.9845--§177.755. 

2.  I  buy  produce  worth  §500,  but  it  will  be  60  days 
before  I  shall  have  money  with  which  to  pay  for  it. 
For  what  sum  must  I  draw,  at  6  ^  per  annum  ? 

Ans.  §505.305. 

3.  I  buy  cotton  to  the  amount  of  §372.93  J,  and  bor- 
row money  at  the  bank  with  which  to  pay  for  it.  For 
what  sum  do  I  draw,  payable  in  30  days,  at  6^  per 
annum?  Ans.  §375. 

PROMISSORY    NOTES. 

247.  A  Promissory  Note  is  a  promise  in  writing  to 
pay  a  certain  sum  of  money  to  a  person,  named  in  the 
note,  or  order,  or  to  the  bearer. 

(1.)  The  Drawer  of  a  note  is  the  person  signing  it. 
(2.)  The  Payee  of  a  note  is  the  person  to  whom  the 
money  is  to  be  paid. 


238  PERCENTAGE. 

(3.)  The  Indorser  of  a  note  is  a  person  who  guaran- 
tees the  payment  of  it.  He  does  this  by  writing  his 
name  on  the  back  of  the  paper  on  which  the  note  is 
written. 

(4.)  An  Indorsement  is  an  acknowledgment  on  the 
back  of  the  note  that,  at  a  given  date,  a  part  of  the 
money  was  paid. 

(5.)  The  Face  of  a  note  is  the  sum  promised  to  be 
paid. 

S48.  To  find  the  amount  due  at  the  maturity  of  a 
note  upon  which  one  or  more  indorsements  have  been 
made, 

I.  When  the  time  of  the  note  is  one  year  or 
less, 

(1.)  Find  the  amount  of  the  face  of  the  note  from  its 
date  to  the  time  of  maturity/. 

(2.)  Find  the  amount  of  each  payment  from  the  time  it 
was  made  till  the  time  the  note  matures. 

(3.)  Subtract  the  sum  of  the  amounts  of  all  the  pay- 
ments from  the  amount  of  the  face  of  the  note.  (Vide 
233,  (2.) 

EXAMPLES. 

$500.  Richmond,  Jan.  1,  1865. 

(1.)  Ninety  days  after  date,  I  promise  to  pay  to  the 
order  of  Frank  H.  Ransom  Five  Hundred  Dollars, 
with  interest,  value  received. 

John  M.  Sabin. 

Indorsements  :  January  20,  $100;  February  10,  $50; 
February  25,  $100;  March  1,  $150. 

What  was  due  at  maturity  ? 


PERCENTAGE. 

239 

OPERATION. 

A^mottnt  of  JJ500  for  93  days  is 

§507.75 

"          §100   "  74      " 

§101.231 

"          §50     ''  53       " 

50.44^ 

§100   "  38       " 

100.631 

"          §150   "  34      " 

150.85 

Sum  of  the  amounts  of  payments, 

.§403.15§ 

Sum  due  at  maturity,  April  4,  1865,  §104.59^ 

II.  When  the  time  of  the  note  is  more  than  1  year, 
(1.)  Find  the  amount  of  the  face  of  the  note  to  the  date 

of  the  first  payment,  and  deduct  the  payment. 

(2.)  Find  the  amount  of  the  remainder  to   the  date 

of  the  next  payment,  and,  after  deducting  the  payment, 

find   the  amount  of  the  remainder  to  the  date  of  the 

next   indorsement,  and   so    on  till   the  last  payment   is 

reached. 

(3.)  Find  the  amount  of  the  remainder,  on  deducting 

the  last  payment,  from  the  date  of  that  payment  till  the 

time  the  note  matures. 

Remark. — Unless  a  payment  or  payments  are  eqnal  to  or  exceed 

the  interest  due  at  the  date  of  the  last,  they  must  be  added  to  the 

succeeding  payment,  and  the  sum  considered  as  a  single  payment. 

In  business,  therefore,  no  payment  should  be  made  on  a  note  unless 

it  exceeds  th«  interest  then  due. 

§450.  Mobile,  Jan.  1,  1865. 

(2.)  Two  years  after  date,  I  promise  to  pay  to  the  order 
of  James  Boone  Four  Hundred  and  Fifty  Dollars,  with 
interest,  value  received.  David  Miller. 

Indorsements:  March  16,  1865,  §75;  January  1, 
1866,  SlOO;    April  4,  1866,  §200. 

What  was  due  at  maturity? 


240  PERCENTAGE. 


OPERATION. 


Amount  of  $450  to  March  16,  is  (236,  Ex.  41.)  |457.50 

75.00 


Payment  deducted  is 382.50 

24.225 


Amount  of  $382.50  to  Jan.  1,  is  (236,  Ex.  42,)  406.725 

100.000 


Pa^^ment  deducted  is 306.725 

6.339 


Amount  of  $306,725  to  April  4,  is  (236,  Ex.  43,)  313.064 

200.000 


Payment  deducted  is 113.064 

6.708 


Amount  of  $113,064  to  Jan.  1,  '62,  (236,  Ex.  44)  $119,772 
$700.  Louisville,  Jan.  1,  1865. 

(3.)  Two  years  after  date,  for  value  received,  I 
promise  to  pay  A.  B.,  or  order,  Seven  Hundred  Dollars, 
with  interest.  M.  Greene. 

Indorsements:  July  28,  1865,  $100;  April  4,  1866, 
$500. 

What  was  due  at  maturity?  (Vide  236,  Ex.  45,  46, 
47.)  Ans.  $156.40. 

$3000.  San  Francisco,  Jan.  1, 1865. 

(4.)  Two  years  after  date,  for  value  received,  I  promise 
to  pay  James  Monroe,  or  order,  Three  Thousand  Dol- 
lars, .with  interest. 

Indorsements:  April  1,  1865,  $1000;  January  1, 
1866,  $1000. 

What  was  due  at  maturity?  Ans.  $1353.68|. 


PEKCEXTAGE.  241 

S.SOO.  Mobile,  June  10,  1865. 

(5.)  June  2,  1866,  for  value  received,  I  promise  to 
pay  S.  S.  Bryant,  or  order,  Three  Hundred  Dollars,  with 
interest.  P.  Hamilton. 

Indorsements :  January  20,  1866,  §116 ;  March  2, 
1866,  §49.50;  April  26,  1866,  §85. 

What  was  due  at  maturity  by  both  rules  ? 

Ans.  I.  $67.89;  II.  §68.17. 
§1000.  Galveston,  Jan.  1,  1869. 

(6.)  July  1, 1870,  for  value  received,  I  promise  to  pay 
C.  Q.  M.,  or  order.  One  Thousand  Dollars,  with  interest. 

Wm.  Daniel. 

Indorsements:  July  1, 1869,  §30;  Jan.  1, 1870,  §470 

What  was  due  at  maturity  by  both  rules? 

Ans.  11.  §603.20  ;  I.  §598.80. 
§400.  Buffalo,  Jan.  1,  1870. 

(7.)  One  year  after  date,  for  value  received,  I  promise 
to  pay  N.  Stacy,  or  order,  Four  Hundred  Dollars,  with 
interest.  M.  M.  DeYoung. 

Indorsements:  March  16,  1870,  §200;  July  1,  §100. 

What  was  due  at  maturity  ? 

Ans.  II.  §113.88 ;  I.  §112.25. 

COMPOUND   INTEREST. 

249.  Compound  Interest  is  interest  on  the  principal, 
and  then,  after  the  interest  becomes  due,  on  the  amount. 
(Vide  233,  (2.) 

examples. 
1.  What  is  the  amount  of  §1.00  at  compound  interest, 
for3yr.,  atS^perannum?  6%?  7%?  8%?  10%? 
21 


242 


PEllCENTAGE. 


OPERATIONS. 

$1.05X1.05X1.05=11.157625  Ans. 
$1.06Xl.06Xl.06-=:|1.191016  Ans. 
$1.07Xl.07Xl.07=$1.225043  Ans. 
$1.08Xl.08Xl.08==$1.259712  Ans. 
$1.10X1.10X1.10=$1.331      (234,  Ex.  5,  G.) 

by   suljtracting 


Remark. — The  compound  interest  is  obtained 
$1  from  the  amounts. 


TABLE, 

Showing  the  amount  of  §1.00  at  compound  interest,  for  any  number 
of  years  from  1  to  25,  at  5,  6,   7,  8,  and  10  per  cent. 


Yeaks 

5  Per  Cent. 

G  Per  Cent. 

7  Per  Cent. 

8  Per  Cent. 

10  Per  Cent. 

"ijMooo" 

1 

1.050000 

1.060000 

1.0700U0 

1.080000 

2 

1.102500 

1.123600 

1.144900 

1.166400 

1.210000 

3 

1.157625 

1.191016 

1.225043 

1.259712 

1.331000 

4 

1.215506 

1.262477 

1.310796 

1.360488 

1.464100 

5 

1.276282 

1.338226 

1.402551 

1.469328 

1.610510 

6 

1.340096 

1.418519 

1.500730 

1.586874 

1.771561 

7 

1.407100 

1.503630 

1.605781 

1.713824 

1.948717 

8 

1.477455 

1.593848 

1.718186 

1.850930 

2.143589 

9 

1.551328 

1.689479 

1.838459 

1.999004 

2.357948 

10 

1.628895 

1.790848 

1.967151 

2.158924 

2.593742 

11 

1.710339 

1.898299 

2.104851 

2.331638 

2.853117 

12 

1.795856 

2.012196 

2.252191 

2.518170 

3.138428 

13 

1.885649 

2.132928 

2.409845 

2.719623 

3.452271 

14 

1.979932 

2.260904 

2.578534 

2.937193 

3.797498 

15 

2.078928 

2.396558 

2.759031 

3.172169 

4.177248 

16 

2.182875 

2.540352 

2.952163 

3.425942 

4.594973 

17 

2.292018 

2.692773 

3.158815 

3.700018 

5.054470 

18 

2.406619 

2.854339 

3.379932 

3.996019 

5.559917 

19 

2.526950 

3.025600 

3.616527 

4.315701 

6.115909 

20 

2.653298 

3.207135 

3.869084 

4.660957 

6.727500 

21  ■ 

2.785963 

3.399564 

4.140562 

5.033834 

7.400250 

22 

2.925261 

3.603537 

4.430402 

5.436540 

8.140275 

23 

8.071524 

3.819750 

4.740530 

5.871404 

8.954302 

24 

8.225100 

4.048935 

5.072367 

6.341181 

9.849733 

25 

3.386354 

4.291871 

5.427433 

6.848475 

10.834700 

PERCENTAGE.  243 

S50.  To  find  the  amount  at  compound  interest  of 
any  principal,  at  any  rate  ^  per  annum,  and  for  any 
given  time, 

(1.)  For  the  given  integral  number  of  years,  multiply 
the  principal  hy  the  amount  of  |1.00,  for  the  same  time 
and  rate   %. 

(2.)  0)1  this  amount  find  the  amount  for  the  months 
and  days,  as  in  §236. 

Remark. — The  compound  interest  will  be  the  diflference  between 
the  amount  and  the  given  principal. 

EXAMPLES. 

1.  Find  the  compound  interest  of  §400  for  9  yr.,  at 
bfo  per  annum.     §400Xl.551328=f620.5312. 

Ans.  $220.53. 

2.  Find  the  compound  interest  of  $400  for  9  yr.  4 mo. 
6  da.,  at  5%  per  annum.     (Vide  236,  Ex.  51.) 

Ans.  $231.39. 

3.  Find  the  compound  interest  for  $1000  for  2  yr. 
8  mo.  15  da.,  at  6  %  per  annum.     (Vide  236,  Ex.  52.) 

Ans.  $171,353. 

4.  Find  the  compound  interest  of  $1250  for  3yr.  3  mo. 
24  da.,  at  7%  per  annum.     (Vide  236,  Ex.  53.) 

Ans.  $315.24. 

5.  Find  the  compound  interest  of  $700  for  20  yr. 
5  mo.  27  da.,  at  10  %  per  annum.     (Vide  236,  Ex.  54.) 

Ans.  $4240.788. 


244  n.vTio. 


RATIO. 


251.  Ratio  is  the  quotient  obtained  by  dividing  one 
number  by  another  of  the  same  kind.     Thus, 

The  ratio  of  5  to  15  is  '^f=^,  commonly  expressed 
by  5  :  15. 

(1.)  The  two  numbers  forming  a  ratio  are  together 
called  terms. 

(2.)  The  first  term  is  called  the  antecedent. 

(3.)  The  second  term  is  called  the  consequent. 

(4.)  The  antecedent  and  consequent  form,  a  couplet. 

(5.)  The  value  of  a  ratio  is  the  quotient  of  the  conse- 
quent divided  hy  the  antecedent. 

(G.)  A  ratio  is  in  its  simjjlest  or  loivcst  terms  -when  the 
terms  are  integral  and  prhyie  tvith  resjject  to  each  other. 
(Vide  102.) 

252.  To  reduce  a  ratio  to  its  lowest  terms, 

(1.)  If  fractions  are  involved,  multijjlt/  the  terms  hy  the 
least  common  multiple  of  the  denominators  of  the  frac- 
tions.    (Yide  107.) 

(2.)  Divide  the  resulting  terms  hy  their  greatest  common 
divisor^  (vide  104,)  or  cancel  such  factors  as  are  common 
to  hoth  terms, 

EXAMPLES. 

1.  Reduce  15:20;  14:21;  16:24  to  their  lowest 
terms,  and  find  their  values.  Values  !{  ;  li ;  IJ. 

2.  Reduce  9:63;  26:169;  34:187  to  their  lowest 
terms,  and  find  their  values.  Values  1\  6^;  5^. 


RATIO. 


245 


3.  Reduce  5|  :  4f ;  2i :  If ;  and  Jf  :  3i  to  their  lowest 
terms,  and  find  their  values. 


OPERATIONS. 

(2.) 
21  :  If 


5|:4f 

119  :  102  68  :  48 

7  :  6  Value  f 

4'.  Reduce  |  :  I ;   j 
and  find  their  values. 


(3.) 


21  :  16  Value  J  f , 


M:3i 
52  :  351 
4 


27  Value  6^, 


1  . 

4  ' 


2   .     7 
6    •    1  Oi 


to  their  lowest  terms, 
Values  i;  f;  If. 


5.   Reduce  4i:5i;    6j:7i;    and  4^  :  2|f    to  their 
lowest  terms,  and  find  their  values. 


VahcesV^;  Ig^;  and  |. 

6. 

Reduce  3i 

.92  .  4.1 

:f;  and  J: 

:  7^  to  their  lowest 

term 

s,  etc. 

Values 

3  ?    T6  ?    ^"^  ^g- 

7. 

Reduce  5  : 

2.5;  .3:21;  and  1/^ 

^  :  3.4  to  their  low- 

est  terms,  etc. 

OPERATIONS. 

(1.) 

(2.) 

(3.) 

5; 

:2.5 

.3:21 

l/,:3.4 

50: 

:25 

3  :  210 

17:34 

2  ; 

:  1    Value  J. 

1:70 

Value  70.  ' 

1:2    Vahie2. 

8. 

Reduce  .5  : 

.2;  Mi: 

.25 ;  and  .4 

:  .7  to  their  lowest 

terms,  etc. 

Values,  Ex.  4. 

9. 

What  is  the  ratio  of  50  cts.  :  20  cts.?  33 1  cts. :  25 

cts.? 

40  :  70  cts. 

? 

Ans.  f;  I;  If. 

10.  What  is  the  ratio  of  14  hu.  :  35  bu.?  2  qt. :  8  qt.? 
30  sec. :  50  sec?  Ajis.  2^-;  4;  f. 

11.  What  is  the  ratio  of  2  qt.  :  3  pk.?  30  sec.  :7m.? 
lpt.:l  gal.?— (Vide  251.) 


246  RATIO. 

OPERATIONS. 
(1.)  (2.)  (3.) 

2  qt. :  3  pk,  30  sec.  :7  m.  1  pt. :  1  gal. 

2  qt. :  24  qt.  30  sec. :  420  sec.  1  pt. :  4  qt. 

1  qt. :  12  qt.   Value  12.    1  sec. :  14  sec.   Value  14.  1  pt. :  8  pt.   Value  8. 

12.  What  is  the  ratio  of  1  mile  to  5  fur.  3  r.  10  ft 
6  in.?— (193,  Ex.  2.)  Ans.  j\. 

13.  What  is  the  ratio  of  1  mile  to  3  fur.  22  r.  3  ft. 
8in.?— (193,  Ex.  4.)  Ans.^^. 

PROPORTION. 

353.  A  Proportion  is  an  equality/  of  ratios. 

The  equality  is  indicated  by  four  dots  or  a  double 
colon  written  between  the  couplets.     Thus, 

5: 15::  6: 18 
is  a  proportion,  and  is  read  5  is  to  15  as  6  is  to  18,  the 
meaning  of  which  is  that  15-^-5  is  the  same  as  18-^-6; 
that  is,  M=:ig^ 

(1.)  The  first  and  last  terms  of  a  proportion  are  called 
extremes. 

(2.)  The  second  and  third  terms  are  called  means. 

(3.)  The  first  and  second  terms  form  the  first  couplet. 

(4.)  The  third  '  and  fourth  terms  form  the  second 
couplet.     Thus,  in  the  proportion  above, 

5  and  18  are  the  extremes;  15  and  6  the  means;  5 
and  15  the  first  couplet;  6  and  18  the  second  couplet; 
5  and  6  the  two  antecedents,  and  15  and  18  the  tAvo 
consequents. 

!354.  Proposition. — If  four  numbers  are  in  propor- 
tion, the  product  of  the  extremes' is  equal  to  the  product 
of  the  means.     Thus, 


RATIO.  247 

From  any  proportion  as  3  :  9  : :  7  :  21, 
we  have  by  251  and  253,  |=V)  ^^^  ^J  multiply- 

ing both  of  these  fractions  by  the  least  common  multiple 
of  their  denominators  we  have  9x7=3x21,  and  it  is 
evident  that  any  proportion  may  be  treated  in  a  similar 
way. 

Remark. — In  any  proportion,  if  the  second  term  is  less  than  the 
first,  the  fourth  will  be  less  than  the  third,  and  if  the  second  term  is 
greater  than  the  first,  the  fourth  will  be  greater  than  the  third. 

S55.  Proposition. — If  in  any  j^'^oportion  the  terms 
of  either  couplet  he  multiplied  or  divided  hy  the  same 
number,  the  proportion  luill  not  he  destroyed.  Thus,  from 
the  proportion, 

7  :  21  : :  8  :  24  we  have 

1  :  3 : : 8  :  24  or  7  :  21 : : 1  :  3 

1^256.   Proposition. — If,  in  any  proportion,   the  two 

antecedents  or  the  two  consequents  he  midtiplied  or  divided 

hy  the  same  numher,  the  proportion  will  not  he  destroyed. 

Thus :  From  the  proportion 

7  :  8  : :  21  :  24  we  have 

1  :  8  : : 3  :  24  or  7  : 1  : :  21 :  3 

257.  Problem. — The  two  extremes  of  a  proportion 
and  one  mean  being  given,  to  find  the  other  mean, 

(1.)  Reduce  the  given  terms  as'loiv  as  possible  hy  255 
and  256. 

(2.)  Divide  the  product  of  the  resulting  extremes  hy  the 
mean.     The  quotient  will  be  the  other  mean. 

EXAMPLES. 

1.  Given  the  terms  7  :  21  : :  :  24,  to  find  the  other 
mean. 


248 


RATIO. 
OPERATION. 


Divide  first  couplet  by  7.     7  :  21 

Divide  consequents  by  3.     1:3 

1:    1 

2.  Given  the  terms  7  : 


mean. 


:24 

:  24  (Vide  255.) 

:    8     Ans.     (Vide  256.) 

8  :  24,  to  find  the  other 
Ans.  21. 


3.  Given,  10  :  14  :  : 


:  35,  to  find  the  other  mean. 


Ans.  25. 

4.  Given,  10  :       : :  25  :  35,  to  find  the  other  mean. 

Ans. 

5.  Given,  85  :  102  : :        :  306,  to  find  the  other  mean. 

A71S.  255. 

6.  Given,  J  :  |  : :       :  f ,  to  find  the  other  mean. 


OPERATION. 

Multiply  first  couplet  by  6.     A  :  | 

Divide  consequents  by  2.         3:4 

3:2 


-I     (Vide  1671  Ex.  106.) 

f     (Vide  255.) 

I     (Vide  256.)  Ans.  j^^. 


7.  Given,  J  :     : :  ^^^  :  §,  to  find  the  other  mean. 


8.  Given,  -/g  :  /,- 


9.  Given,  f  :  24  : 


Ans.  f . 
:  100,  to  find  the  other  mean. 
Ans.  120. 
:  20,  to  find  the  other  mean. 
Ans.  j\. 
10.  Given,  |  :  ^5_  . .       .  20,  to  find  the  other  mean. 

258.  The  two  means  of  a  proportion  and  one  extreme 
being  given  to  find  the  other  extreme, 

(1.)  Reduce  the  terms  as  low  as  possible  by  255  and 
256. 

(2.)  Divide  the  product  of  the  resulting  means  by  the 
extreme.     The  quotient  will  be  the  other  extreme. 


RATIO.  249 

EXAMPLES. 

1.  Given  of  :  4f  : :  21,  to  find  the  fourth  term. 

OPERATION. 

Multiply  couplet  by  21.     5|  :    4f  : :  21  (Vide  252, Ex.  3.) 
Divide  couplet  by  17.     119  :  102  : :  21  (Vide  255.) 
Divide  antecedents  by  7.      7  :      6  : :  21  (Vide  256.) 

1  :      6  : :    3  :  18  Ans. 

2.  Given,  1 :  2  : :  3,  to  find  the  fourth  term.     A7is.  6. 

3.  Given,  3:9::  12,  to  find  the  fourth  term. 

Ans.  36. 

4.  Given,  4  :  16  r :  15,  to  find  the  fourth  term. 

Ans,  60. 

5.  Given,  ^%  :  /o  : :  120,  to  find  the  fourth  term. 

Ans.  100. 

6.  Given,  1  :  |  : :  J,  to  find  the  fourth  term. 

Ans.  I. 

7.  Given,  9  :  18  : :  ^,  to  find  the  fourth  term. 

A71S.  1. 

8.  Given,  35|  :  15^  : :  4,  to  find  the  fourth  term. 

Ans.  IjV^. 

9.  Given,  4^  :  22^  : :  9,  to  find  the  fourth  term. 

Ans.  45. 

10.  Given,  ^^  :  28?  : :  3,  to  find  the  fourth  term. 

Ans.  27. 

11.  Given,  2.5  :  45  : :  63,  to  find  the  fourth  term. 

A71S.  1134. 

12.  Given,  4.5  :  22.5  : :  9,  to  find  the  fourth  term. 

A71S.  45. 

13.  Given,  J-i  :  ^J-  : :  ^%,  to  find  the  fourth  term. 

Ans.  §. 


250  RATIO. 

14.  Given,  5  :  f  : :  |,  to  find  the  fourth  term. 

Ans.  /j. 

15.  Given  7|  :  6  : :  5|,  to  find  the  fourth  term. 

Ans.  4j|-f. 

16.  Given,  6  :  7|  : :  5f ,  to  find  the  fourth  term. 

Ans.  6§§. 

17.  Given,  1  :  2|  : :  2|,  to  find  the  fourth  term. 

Ans.  6i. 

18.  Given,  75|  :  36  : :  643|,  to  find  the  fourth  term. 

Ans.  306. 

19.  Given,  1.50  : 1.00  : :  .30,  to  find  the  fourth  term. 

Ans.  .20. 

20.  Given,  3X32  :  5X16  ::  120,  to  find  the  fourth 
term.  Ans.  100. 

21.  Given,  8X21  :  56X6  ::  10,  to  find  the  fourth 
term.  Ans.  20. 

22.  Given,  2\  :  If  : ;  |f ,  to  find  the  fourth  term. 

Ans.  ||. 

23.  Given,  •     :^%:'.  120  :  100,  to  find  the  first  term. 

A71S.  /g. 

RULE    OF    THREE. 

259.  Every  problem  in  proportion  involves  at  least 
three  quantities,  so  related  to  each  other  that  a  fourth 
may  be  found  from  them. 

260.  The  statement  of  a  problem  consists  in  properly 
arranging  the  three  quantities  mentioned  in  it,  so  as  to 
form  the  first,  second,  'dud  third  terms  of  a  proportion. 

261.  The  Rule  of  Three  consists  of  directions  by 
which  a  problem  in  proportion  may  be  stated.  They 
are  as  follows : 


HATIO.  251 

(1.)  Of  the  three  quantities  mentioned,  make  that  the 

THIRD    TERM   wMcJl   JiaS    the    SAME   NAME    aS    the    ANSWER 

required. 

(2.)  Of  the  two  remaining  quantities,  make  the 
GREATER  the  SECOND  TERM  if  the  ANSWER  ought  to  be 
GREATER  than  the  third  term;  make  the  less  the  sec- 
ond term  if  the  answer  ought  to  be  less  tha7i  the  third 
TERM. — (Vide  254,  Rem.) 

(3.)  Place  the  remaining  quantity  for  the  first  term. 

The  fourth  term,  found  by  258,  will  be  the  answer, 

examples. 

1.  If  5f  lb.  of  sugar  cost  21  cts„  what  will  4!^  lb.  cost? 
(1.)  21  cts.  has  the   same   name  which  the   answer 

should  have. 

(2.)  4|  lb.  will  evidently  cost  less  than  21  cts.,  the 
jorice  of  5f  lb. 

(5.)  5|  should  then  be  the  first  term,  4?  the  second, 
and  21  cts.  the  third. 

Thus    5§  :  4^  : :  21  (vide  258,  Ex.  1.)     Ans.  18  cts. 

2.  If  j\  of  a  quantity  of  sugar  cost  S120,  what  will 
/j  of  the  same  quantity  cost?— (See  167 J,  Ex.  113, 
Rem.;  and  258,  Ex.  5.)  Ans.  |100. 

3.  If  3  yd.  of  cloth  cost  $12,  what  will  9  yd.  cost? 

A71S.  §36. 

4.  If  4  lb.  of  rice  cost  15  cts.,  what  will  16  lb.  cost? 

Ans.  60  cts. 

5.  If  1  lb.  of  tea  cost  J  of  a  dollar,  what  will  J  lb. 
cost?  Ans.  16|  cts. 

6.  If  a  staff  9  feet  high  cast  a  shadow  \  of  A  foot 


-OJ  RATIO. 

LONG,  what  will  be  the  length  of  the  shadow  of  a  post 
which  is  l^feet  high  f  Ans.  258,  Ex.  7. 

7.  If  a  tree  35|  ft.  high  casts  a  shadow  4  ft.  long, 
what  will  be  the  length  of  the  shadow  of  a  tree  15 1  ft. 
Mgh  1  Am.  1  ^V?  ^.=1  ft.  .8  ^V^  in. 

8.  If  a  staff  10  ft.  high  casts  a  shadow  12  ft.  in  lengthy 
what  will  be  the  hight  of  a  tree  whose  shadow  measures 
70  ft.  ?  Am.  58  J  ft.=58  ft.  4  in. 

9.  If  34  yd.  of  cloth  cost  ^3,  what  will  28|  yd.  cost? 

Am.  |27. 

10.  If  2.5  A.  of  land  cost  |63,  what  will  45  A.  cost? 

Am.  §1134. 

11.  If  45  A.  of  land  cost'  |1134,  what  will  2.5  A. 
cost  ?  Am. 

12.  If  45  A.  of  land  cost  §1134,  what  quantity  can 
be  bought  for  §63  ?         1134  :  63  : :  45  Am. 

13.  If  4.5  yd.  of  cloth  cost  §9,  what  will  22.5  yd. 
cost  ? 

14.  If  25.5  yd.  of  cloth  cost  §45,  what  will  4.5  yd. 
cost? 

15.  If  J  J  of  a  pound  of  butter  cost  -^^  of  a  dollar, 
how  much  will  ij  of  a  pound  cost? 

16.  If  W  of  a  lb.  of  butter  cost  40  cts.,  what  will 
jj  lb.  cost? 

17.  If  \\  of  a  lb.  of  butter  cost  53 1  cts.,  what  quan- 
tity can  be  bought  for  40  cts.?  An8.  8|  oz. 

18.  If  8  J  oz.  of  butter  can  be  bought  for  40  cts.,  what 
quantity  can  be  purchased  for  53  J  cts. 

An%.  11J4  oz. 

19.  If  6  yd.  of  cloth  cost  §4.55||,  what  will  71  yd. 
cost  ? 


RATIO.  253 

20.  If  75  yd.  of  cloth  cost  ^5.60,  Avhat  will  G  yd. 
cost? 

21.  If  1  yd.  of  carpeting  cost  |2-J,  what  will  2 J  yd. 
cost? 

22.  If  2  J  yd.  carpeting  cost  ^6.25,  what  will  .one  yd. 
cost? 

23.  If  75f  yd.  of  cloth  cost  |643|,  what  will  3G  yd. 
cost?— (Vide  167i,  Ex.  72;  and  258,  Ex.  18.) 

24.  If  1  coat  require  If  yd.  of  cloth,  how  many  coats 
can  be  made  of  18 J  yd.?  Ans.  10  coats. 

25.  If  10  lb.  of  copper  cost  ^18.75,  what  number  of 
pounds  can  be  had  for  ^171.25?— (Vide  167J,  Ex.  79 
and  80.) 

26.  If  f  of  a  ship  are  worth  $15700,^  what  is  the 
value  of  i  of  the  ship?  •  Ans.  167},  Ex.  85. 

27.  If  1  lb.  of  butter  cost  3  pence,  how  many  tons 
can  be  bought  for  84  £  13  s.  9d.? 

Ans.  191,  Ex.  63. 

28.  If  I  buy  cloth  at  §1.50  and  sell  it  for  §1.20, 
what  shoiild  I  lose  on  §1.00?— (Vide  258,  Ex.  19.) 

Ans.  §0.20. 

29.  If  I  buy  cloth  at  §1.25  and  sell  it  for  §1.75, 
what  should  I  make  on  §1.00?— (Vide  228,  Ex.  8.) 

Ans.  40  cts. 

30.  If  I  pay  at  the  rate  of  11  cts.  per  mile,  how  far 
can  I  ride  for  7  cts.?  Ans.  193,  Ex.  39. 

31.  If  I  sell  a  quantity  of  land  at  the  rate  of  §450 
for  25  sq.  r.,  and  obtain  §51300,  how  many  acres  do  I 
sell?  Ans.  191,  Ex.  74:. 

32.  If  2.5  lb.  of  tobacco  cost  75  cts.,  how  much  will 
185  pounds  cost?  Ans.  §55.50. 


254  RATIO. 

33.  If  2  oz.  of  silver  cost  ^2M,  wliat  will  f  oz.  cost? 

Ans.  $0.84. 

34.  If  7  lb.  of  sugar  cost  75  cts.,  what  will  12  pounds 
cost?  Ans.  $1,284. 

35.  If  141  tons  of  coal  cost  85  £,  what  will  94  tons 
cost?  Ans.  $274.26. 

36.  If  the  interest  of  $19.35  is  $2.7477,  what  will  be 
the  interest  of  $17.21  for  the  same  time  and  rate  per 
cent.?  Ar^s.  $2.44382. 

37.  If  the  interest  of  $17.21  is  $2.63313,  what  will 
be  the  interest  of  $140.10  for  the  same  time  and  rate 
per  cent.?  Ans.  $21.4353. 

38.  If  4  men  can  do  a  piece  of  work  in  100  days,  in 
how  many  days  would  5  men  do  the  same  work? 

Ans.  80  days. 

39.  If  20  men  can  do  a  piece  of  work  in  J  of  a  day, 
how^  long  would  it  take  2  men  to  do  the  same  w^ork? 

Ans.  5  days. 

40.  If  60  men  can  do  a  piece  of  work  in  8  days,  how 
many  men  would  perform  the  same  w^ork  in  20  ^ays? 

Ans.  24  men. 

41.  If  2  men  cai>  dig  a  ditch  in  40  days,  in  how  many 
days  would  10  men  dig  the  same  ditch? 

A71S.  8  days. 

42.  Having  read  120  pages  of  a  book,  I  find  that  I 
have  still  to  read  |  of  the  book.  How  many  pages  does 
the  book  contain? 

§  :  I  : :  120  Ans.  300  pages. 

43.  A  person  owning  |  of  a  coal  mine  sells  |  of  his 
share  for  $400.     What  is  the  mine  worth  at  this  rate? 

Ans.  $1000. 


iiATio.  255 

44.  If  10  covfs  eat  8  tons  of  hay  in  a  given  time, 
how  many  cows  would  eat  56  tons  in  the  same  time? 

Ans.  70  cows. 

45.  If  70  cows  eat  a  certain  quantity  of  hay  in  6 
weeks,  how  many  coavs  would  eat  the  same  hay  in  21 
weeks?  Ans.  20  cows. 

46.  If  20  men  have  been  at  work  18  days  to  con- 
struct a  given  length  of  railroad,  how  many  days  will 
76  men  require  to  construct  the  same  length  of  road  ? 

Ans.  4i|  days. 

47.  If,  by  making  10  hours  a  day,  a  certain  number 
of  men  complete  a  work  in  18  days,  how  many  days 
would  be  required  to  complete  a  similar  work,  at  12 
hours  a  day?  Ans.  15  days. 

48.  If  18  days  are  required  to  construct  500  feet  of 
railroad,  how  many  days  will  be  occupied  on  1140  feet 
of  the  same  road?  Aiis.  41^^  days. 

49.  If  a  passenger  train  of  cars  gain  on  a  freight 
train  at  the  rate  of  8  miles  in  3  hours,  how  many  hours 
will  it  take  to  gain  60  miles?  Ans.  22^  hours. 

50.  A  passenger  train  of  cars  moves  at  the  rate  of 
45  miles  in  3  hours,  and  a  freight  train  at  the  rate  of 
37  miles  in  the  same  time.  If  the  freight  train  has  60 
miles  the  start,  in  what  time  will  it  be  overtaken  ? 

51.  A  hand  car,  running  at  the  rate  of  only  1  mile 
an  hour,  has  12  miles  the  start  of  a  passenger  train, 
which  runs  at  the  rate  of  12  miles  an  hour.  In  what 
time  will  the  hand  car  be  overtaken? 

Ans.  1  h.  5y\  m. 

52.  If  the  long  hand  of  a  clock  move  at  the  rate  of 
12  spaces  an  hour,  and  the  short  hand  1  sjmce  an  hour, 


256  RATIO. 

in  what  time  will  the  long  hand  gain  12  spaces  upon  the 
short  hand? 

53.  At  what  time  between  9  and  10  o'clock  will  the 
hands  of  a  clock  be  together? 

Ans.  9  o'clock,  49  Jj-  m. 

S62.  To  state  a  j)roblem  when  it  involves  more  than 
three  terms, 

(1.)  Of  the  quantities  mentioned,  maJce  that  the  third 
term  which  has  the  same  name  as  the  answer  required. 

(2.)  Select  tivo  terms  of  the  same  name,  and  arrange 
the  couplet  as  though  it  were  entirely  disconnected  with  all 
other  conditions  of  the  i^rohlem.     (Vide  261,  (2.) 

(3.)  Select  two  other  terms  of  the  same  name,  and  ar- 
range as  before,  and  so  on,  till  all  the  teyms  are  arranged. 

Having  canceled  all  the  factors  common  to  any  ante- 
cedent and  consequent,  (vide  255,)  or  common  to  either 
one  of  the  first  terms  and  the  third,  (vide  256,)  the  co7i- 
tinued  product  of  the  means  divided  hy  that  of  the  first 
terms  ivill  he  the  ansiuer. 

EXAMPLES. 

1.  If  10  cows  eat  8  tons  of  hay  in  6  weeks,  how  many 
cows  will  eat  56  tons  in  21  weeks? 

It  would  take  more  cows  to  eat  56  tons  than  8  tons, 
and  less  cows  to  eat  it  in  21  weeks  than  6  weeks. 

STATEMENT. 

(Vide  261,  Ex.  44  and  45.)  8 :  66      .^        (vide  258,  Ex.  21.) 

21:    6"^" 

SOLUTION. 
Divide  upper  couplet  by  8.       1-7..;^q  (vide  also  156,  Ex.  22.) 

Divide  lower  couplet  by  3  and  cancel  the  7s.  Ans.  20  cows. 


RATIO.  257 

2.  If  the  wages  of  6  men  for  14  days  be  §84,  what 
will  be  the  wages  of  9  men  for  16  days?    Ans.  ^144. 

3.  If  12  oz.  of  wool  make  21  yd.  of  cloth  which  is 
1^  yd.  zvide,  how  many  pounds  will  it  take  to  make 
150  yd.  only  1  yd.  wide?  Ans.  30  lb. 

4.  If  the  interest  of  §19.35  for  1  yr.  2  mo.  6  c!a.  is 
§2.7477,  what  is  the  interest  of  §17.21  for  1  yr.  3  mo. 
9  da.,  the  rate  per  cent,  being  the  same  in  each  case  ? 
(Vide  205,  Ex.  56;  also  236,  Ex.  3.)     Ans.  §2.63313. 

5.  If  the  interest  of  §140.10  for  2  yr.  4  mo.  12  da. 
is  §39.7884,  what  is  the  interest  of  §75.15  for  3  yr. 
6  mo.  15  da.,  the  rate  per  cent,  being  12  in  each  case? 
(Vide  205,  Ex.  56 ;  .236,  Ex.  5.)  Ans.  §31.93|. 

6.  How  much  hay  will  32  horses  eat  in  120  days,  if 
96  horses  eat  3|  tons  in  7 J  weeks?         A^is.  2f  tons. 

7.  If  6  laborers  dig  a  ditch  34  yd.  long  in  10  days, 
how  many  yards  will  20  laborers  dig  in  15  days? 

8.  If  20  laborers  dig  a  ditch  17-0  yd.  long  in  15  days, 
how  many  days  will  6  laborers  require  to  dig  a  ditch 
34  yd.  long  ?  Ans. 

9.  If  14  men  can  reap  84  acres  in  6  days,  how  many 
men  must  be  employed  to  reap  44  acres  in  4  days? 

10.  If  20  men,  by  Avorking  10  hours  a  day,  have  been 
employed  18  days  in  constructing  500  ft.  of  railroad, 
how  many  days  of  12  hours  each  must  76  men  bo  em^ 
ployed  to  construct  1140  feet  of  the  same  road. 


STATEMENT. 

(Vide  261,  Ex.  46,  47,  48.)     76  :      20 
(Vide  157,  Ex.  27)  12 

500 

99 


10::  18 
1140     A71S.  9  davs. 


258  RATIO. 

11.  If  50  men,  by  working  5  hours  a  day,  can  «^'>  9A 
cellars  in  54  days,  each  cellar  being  36  ft.  lo- 

wide,  and  10  ft.  deep,  how  many  men  can  dig  lb  ctii..  s 
in  27  days,  each  cellar  being  48  ft.  long,  28  ft.  wide,  and 
9  ft.  deep,  by  working  3  hours  a  day? 

Ans.  200  men. 

12.  If  496  men,  in  5^  days  of  11  hours  each,  dig  a 
trench  of  7  degrees  of  hardness,  465  ft.  long,  3|  ft.  wide, 
2|  ft.  deep,  in  how  many  days,  of  9  hours  long,  will  24 
men  dig  a  trench,  of  4  degrees  of  hardness,  337 i  ft. 
long,  5f  ft.  wide,  and  3 J  ft.  deep? 

Ans.  157,  Ex.  23. 

13.  If  12  men  can  build  a  wall  30  ft.  long,  6  ft.  high, 
and  3  ft.  thick  in  15  days,  when  the  days  are  12  hours 
long,  in  what  time  will  60  men  build  a  wall  300  ft.  long, 
8  ft.  high,  and  6  ft.  thick,  when  they  work  only  8  hours 
a  day?  ^?zs.  157,  Ex.  24. 

14.  If  25  pears  can  be  bought  for  10  lemons,  and  28 
lemons  for  18  pomegranates,  and  1  pomegranate  for  48 
almonds,  and  50  almonds  for  70  chestnuts,  and  108 
chestnuts  for  2 J  cts.,  how  many  pears  can  I  buy  for 
$1.35?  Ans.  375  pears. 

15.  If  the  interest  of  J2187.50,  from  Aug.  10,  1868, 
to  Jan.  1,  1870,  at  8  per  cent,  per  annum,  is  |243.54^, 
what  is  the  interest  of  |10500,  from  Feb.  1,  1867,  to 
May  13,  1869,  at  the  rate  of  5  per  cent,  per  annum? 

STATEMENT. 

218750:1050000 

167:274         : :  243.54J 
8:5 

Ans.  236,  Ex.  27. 


KATIO.  250 

2$IS.  To  divide  a  given  number  into  parts  "which 
shall  be  proportional  to  given  numbers, 

(1.)  If  the  given  numbers  are  fractions,  midfij^Ii/  them 
all  hy  the  least  common  multiple  of  the  denominators. 

(2.)  3Iake  the  sum  of  the  results  the  first  term  of  a 
proportion,  any  one  of  the  residts  the  second  term,  and 
the  given  number  the  third  term. 

The  fourth  term,  found  by  258,  will  be  one  of  the 
required  parts,  and  the  others  may  be  found  in  like 
manner. 

EXAMPLES. 

1.  Divide  49  into  two  parts  which  shall  have  the  ratio 
of  I  •  ^ 


2 


OPERATION. 


iX6=3.       (Vide  167J,  Ex.  106.) 
PROOF.  fx6=4. 

21+28=49.  7  :  3  : :  49  :  21       1st  part. 

21  :  28  : :  J  :  I  7  :  4  : :  49  :  28       2d  part.^ 

2.  Divide  136  into  two  parts  which  shall  be  propor- 
tional to  the  numbers  |  and  f .  (Vide  167J,  Ex.  107.) 

Ans.  64  and  72. 

3.  Divide  544  into  two  parts  which  shall  be  propor- 
tional to  the  numbers  /^  and  ^^.  (Vide  167J,  Ex.  108.) 

Ans.  384  and  160. 

4.  Divide  2135  into  two  parts  which  shall  be  pro- 
portional to  the  numbers  35 §  and  15^.  (Vide  167 J, 
Ex.  109.)  Ans.  1498  and  637. 

5.  Divide  209  into  three  parts  which  shall  be  propor- 
tional to  the  numbers  J,  |-,  and  |. 


260 


PvATIO. 


OPEEATION. 

^.X12=6.     (Vide  167J,  Ex.  110.) 
J  X 12-4. 

fXl2-=9. 

19:  6::  209:66     1st  part. 


PROOF. 

66+44+99=209. 

66  :  44  ::  J  :  J   (vide  254.)     19  :  4  : :  209  :  44     2d  part. 

66  :  99  ::  i  :  I  19  :  9  : :  209  :  99     3d  part. 

6.  Divide  504  into  four  parts  which  shall  be  propor- 
tional to  the  numbers  J,  f ,  j%,  and  f .     (167J,  Ex.  111.) 

Ans.  135;  180;  81;lindl08. 

7.  Divide  30;  45;  75;  135;  180;  and  750,  each  into 
five  parts  which  shall  be  proportional  to  the  numbers  1, 
2,  3,  4,  and  5.  Ans.  2;  4;  6;  8;  10,  etc. 

8.  Divide  36  into  three  parts,  so  that  J  the  first,  |  the 
second,  and  i  the  third,  shall  all  be  equal  to  each  other. 

Remark, — The  parts  will  evidently  be  as  the  numbers  2,  3,  and  4. 

Ans.  8;  12;  and  16. 

9.  Divide  136  into  two  parts,  so  that  -f  of  the  first 
and  I  of  the  second  shall  be  equal. 

Remabk. — The  numbers  will  be  as  ii  :  i,  or  as  9  :  8. 

A71S.  72  and  64. 

10.  A  gold  and  a  silver  watch  together  cost  §132,  and 
the  gold  watch  costs  10  tijnes  as  much  as  the  silver 
watch.     What  did  each  cost?         Ans.  §120  and  $12. 

11.  A's  age  is  double  B's,  and  B's  is  triple  C's.  The 
sum  of  all  their  ages  is  140.     What  is  the  age  of  each? 

Remark. — Their  ages  are  as  the  numbers,  1,  3,  and  G. 

An8.A8i;  B42;  C  14. 

12.  A  gentleman  bought  a  certain  number  of  oxen, 
and  double  the  number  of  cows;  and  also  three  tiiiics 


RATIO.  261 

as  many  sheep  as  cows.     He  gave  ^50  each  for  oxen, 
$25  each  for  coavs,  and  |3  each  for  sheep;  the  whole 
costing  §354.     What  number  of  each  did  he  purchase  ? 
Ans.  3  oxen;  6  cows;  18  sheep. 

13.  A  man  paid  |74  for  a  sheep,  a  cow,  and  an  ox. 
The  cow  was  valued  at  12  sheep,  and  the  ox  two  cows. 
What  was  the  price  of  each? 

Ans.  sheep  §2;  cow  §24;  ox  §48. 

14.  I  wish  to  make  a  mixture  of  360  pounds  of  tea, 
using  at  the  rate  of  30  lb.  worth  30  cts.  a  pound ;  11  lb. 
worth  33  cts.  a  pound ;  23  lb.  worth  67  cts.  a  pound ;  and 
26  lb.  worth  86  cts.  per  pound.  What  quantity  of  each 
kind  must  be  used? 

Ans.  120,  44,  92,  and  104  lbs.,  respectively. 

204.  Partnership  is  an  association  of  two  or  more 
individuals  for  the  transaction  of  business. 

(1.)  The  partners  constitute  the  company,  firm,  or 
house. 

(2.)  The  capital  is  the  money  invested  by  the  company 
in  business. 

(3.)  The  profit  or  loss  to  be  shared  is  called  a  divi- 
dend. 

S05.  To  ascertain  the  dividend  of  the  partners, 
when  the  money  of  each  has  been  invested  the  same 
length  of  time. 

Make  the  capital  of  the  company  the  first  term  of  a 
projwrtion,  the  money  of  any  partner  the  second  term, 
and  the  profit  or  loss  the  third  term. 

The  fourth  term,  found  by  258,  will  be  the  divi- 
dend of  the  partner  whose  money  forms  the  second 
term. 


262  RATIO. 


EXAMPLES 


1.  A  and  B  invest  $550  in  a  speculation,  of  which 
A  furnishes  $330  and  B  $220.  They  gain  §70.  What 
is  the  dividend  of  each? 

OPERATION. 

550  :  330  : :  $70  :  A's  share. 
5:    3    ::$70:$42 
Divide  antecedents  by  110. 

550  :  220  : :  $70  :  B's  share. 
5:    2    ::$70:$28 
(Vide  232,  Ex.  1.)  Ans.  A  $42;  B  $28. 

2.  A  invests  $300-  in  ti  speculation,  and  B  $400. 
They  gain  $49.  What  is  the  share  of  each?  (Vide 
263,  Ex.  1.)  Ans.  A  $21 ;  B  $28. 

3.  A,  B,  and  C  form  a  partnership.  A  furnishes 
$1200,  B  $1600,  and  C  2000.  What  is  each  partner's 
share  in  a  gain  of  $960? 

Alls.  A  $240;  B  $320;  C  $400. 

4.  A,  B,  and  C  form  a  partnership;  A  furnishing 
$800,  B  $1500,  and  C  $3000.  They  gain  $500.  What 
is  the  dividend  of  each? 

Ans.  A  75.471;  B  $141,509;  C  $283.02. 

5.  A  and  B  form  a  partnership.  A  furnishes  $1200, 
and  B  $500.  They  gain  $544.  What  is  the  share  of 
each?  Ans.  263,  Ex.- 3. 

6.  A,  B,  C,  and  D  make  up  a  purse  to  buy  lottery 
tickets.  A  furnishes  $15,  B  $20,  C  $9,  and  D  $12. 
What  is  each  one's  share  in  a  prize  of  $504? 

Ans.  263,  Ex.  6. 

7.  A,  B,  C,  D,  E,  F,  and  G  engage  in  an  oil  specula- 


RATiO.  .    263 

tion.  A  furnislies  §175  of  the  capital,  B  S500,  C  $600, 
D  ,^210,  E  ^42.50,  F  §20,  and  G  §10.  They  expend 
§623  in  prospecting,  and  then  give  the  matter  up  as  a 
total  failure.    What  does  each  lose?    ^ns.  232,  Ex.  2. 

8.  Three  gentlemen  engage  in  a  gold  speculation.  A 
furnishes  §500,  B  §1200,  and  C  §4300.  They  clear 
§1500.    What  is  each  man's  share?    A71S.  232,  Ex.  3. 

!S66.  To  ascertain  the  dividend  of  the  partners,  when 
the  money  of  each  has  been  invested  different  lengths 
of  time, 

(1.)  Multiply  each  jjartner^s  money  hy  the  time  it  is 
invested. 

(2.)  Make  the  swn  of  the  ijroducts  the  first  term  of  a 
'proportion^  any  one  j^rodiict  the  second  term,  and  the  gain 
or  loss  the  third  term. 

The  fourth  term,  found  by  258,  will  be  the  share  of 
the  partner  whose  product  forms  the  second  term. 

EXAMPLES. 

1.  A  and  B  are  associated  in  trade.  A  has  furnished 
of  the  joi7it  stock  §330  for  5  months,  and  B  §220  for  8 
months.  They  gain  §170.50.  What  is  the  share  of 
each? 

OPERATION. 

330X5=1650. 

220X8=1760. 


Divide  the  coup-  3410  :  1650  : :  §170.50 
let  by  10,  and  the  2  :    165  : :      §1 

two  antecedents  by  3410  :  1760  : :  §170.50 
170.1  2:    176::      SI 


A's  share. 
§82.50 
B's  share. 


264  TvATIO. 

2.  A  and  B  enter  into  partnership.  At  first  A,  with 
a  capital  of  §3G0,  does  business  alone.  At  the  expira- 
tion of  two  months  B  comes  in  with  a  capital  of  §520, 
and  the  partners  do  business  together  5  months.  The 
profits  of  the  concern,  from  the  time  A  commenced 
business,  were  $128.    What  should  be  the  share  of  each? 

Ans.  A  $63;  B  $65. 

3.  A,  B,  and  C  form  a  partnership.  "  A's  part  of  the 
capital  is  $4300,  B's  $2000,  and  C's  $1500.  At  the  end 
of  2  months  A  withdrew  with  his  stock.  At  the  end 
of  another  2  months  B  withdrew  with  his  stock.  C  con- 
tinued the  business  alone  for  another  2  months,  when 
the  entire  profits  were  found  to  have  been  $1280.  What 
is  the  share  of  each? 

A71S.  A  $430;  B  $400;  C  $450. 

4.  On  the  1st  of  January,  1866,  A  commenced  busi- 
ness wdth  a  capital  of  $8000.  On  the  1st  of  July  B 
joins  him  with  a  capital  of  $16000.  On  the  1st  of 
July,  1867,  it  is  found  that  $4000  have  been  cleared  since 
A  began  business.     What  is  the  share  of  each? 

Ans.  A  $171^;  B  $2285|. 

5.  A,  B,  and  C,  at  the  end  of  a  partnership,  have 
jointly  $1000  in  trade.  A's  stock  has  been  in  the  busi- 
ness 7  months,  B's  stock  8  months,  and  C's  12  months. 
A's  dividend  is  $21 ;  B's  $40;  C's  24.  What  amount  of 
money  did  each  invest? 

A71S.  A  $300;  B  $500;  C  $200. 

6.  Two  cousins,  George  and  Frank  Ransom,  com- 
menced business  on  the  1st  of  January,  each  partner 
putting  in  $10000.  On  the  1st  of  June  George  in- 
creased  his    stock  by  $2000,  Frank  withdravring   the 


RATIO.  265 

same  sura.  On  the  1st  of  September  George  withdrew 
§4000,  but  Frank  increased  his  by  §3000.  At  the  end 
of  the  year  they  had  made  §6720.  What  was  the  divi- 
dend of  each?       Ans,  George  §3360;  Frank  §3360. 

EQUATION    OF    PAYMENTS. 

267.    To  ascertain  the  mean  time  for  the  payment  of 

several  sums  due  at  different  times, 

(1.)  Multipli/  each  ijayment  by  its  lime  of  credit. 

(2.)  Divide  the  sum  of  the  products  by  the  sum  of  the 
payments.     The  quotient  will  be  the  mean  time. 

EXAMPLES. 

1.  I  owe  a  merchant  §30,  due  in  4  months;  §40,  due 
in  5  months,  and  §50,  due  in  6  months.  What  is  the 
mean  time  for  the  payment  of  all  the  bills  ? 

OPERATION'. 

30X4=120 
40X5=200 
50X6=300 


120     )     620(5J 

Ans.  5^  months. 

Remark. — The  interests  of  the  several  sums,  at  any  rate  per  cent, 
per  annum,  for  the  given  times,  when  added  together,  must  be  the 
same  as  the  interest  of  the  sum  of  the  payments  for  the  mean  time. 
Hence  the  following  rule: 

(1.)  Find  the  interest  of  each  payment  for  its  time  of  credit. 

(2.)  Divide  the  sum  of  the  interests  hy  the  interest  of  the  sum  of  the 
payments  for  1  month.     The  quotient  will  be  the  mean  time. 

Remark. — If  the  rate  of  12  per  cent,  per  annum  is  selected,  the 
operation  will  be  identical  with  that  above. 

23 


266  RATIO. 

2.  A  merchant  owes  §200,  payable  in  4  months ;  $400, 
in  5  months;  |500,  in  6  months;  and  §600,  in  8  months. 
What  is  the  mean  time  of  payment? 

Ans.  6j\  months. 

3.  A  merchant  has  given  three  notes  to  the  same 
creditor  :  §200,  due  in  2  months ;  §200,  in  4  months ; 
and  §200,  in  6  months.  What  is  the  mean  time  of 
payment  ?  Ans.  4  months. 

4.  I  have  several  bills  due  at  a  store;  §40  due  in  20 
days  from  January  1,  1866;  §30  due  in  40  days  from 
the  same  time;  amd  §50  due  in  45  days.  What  is  the 
mean  time  of  payment  ? 

A71S.  35/^  days;  i.  e.  Feb.  4,  1866. 

5.  I  buy  a  house  and  lot  for  §1600,  with  the  under- 
standing that  I  am  to  pay  one  fourth  cloiV7i,  one  third  of 
the  balance  in  3  months,  and  the  remainder  in  6  months. 
What  would  be  the  mean  time  for  the  payment  of  the 
whole  sum  ?  Ans.  3|  months. 

ALLIGATION    MEDIAL. 

268.  To  ascertain  the  mean  price  of  a  compound 
consisting  of  ingredients  of  which  the  quantity  and 
value  of  each  are  given, 

Divide  the  entire  cost  of  the  compound  hy  the  sum  of 
the  ingredients.     The  quotient  will  be  the  mean  price. 

EXAMPLES. 

1.  A  merchant  bought  160  gallons  of  wine  at  40  cts. 
per  gallon ;  75  gallons  at  60  cts.  per  gallon ;  225  gal- 
lons at  48  cts.  per  gallon ;  40  gallons  at  87 J  cts.  per 
gallon,     What  wns  the  mean  cost  of  the  wine? 


RATIO.  267 


OPERATION. 

160X.40  =  64.00 
75X.60  =  45.00 

225X.48  =-108.00 
40X.87J=  35.00 


500       )       252.00(.505 
252.00 
Ans.  10.504  per  gallon. 

2.  A  farmer  mixed  wheat,  viz  :  5  bushels  worth  |1.10 
per  bushel;  10  bushels  worth  60  cts.  per  bushel;  5 
bushels  worth  70  cts.  per  bushel.  What  is  the  mean 
price  of  the  mixture  ?  Ans.  75  cts.  per  bushel. 

3.  A  wine  merchant  mixes  wine,  viz:  88  gallons  of 
Canary,  worth  50  cts.'  per  gallon ;  88  gallons  of  Sherry, 
worth  76  cts.  per  gallon ;  and  48  gallons  of  Claret,  worth 
^1.75  per  gallon.  What  is  a  gallon  of  the  mixture 
w^orth  ?  A71S.  87  cents. 

4.  A  goldsmith  mixes  7  ounces  of  gold  23  carats  fine 
with  3  ounces  16  carats  fine,  3  oz.  of  18,  and  3  of  19 
carats  fine.     What  is  the  fineness  of  the  mixture  ? 

Ans.  20  carats. 

5.  If  I  mix  30  lb.  of  tea  at  30  cts.  per  lb.;  11  lb.  at 
33  cts.  per  lb. ;  23  lb.  at  67  cts.  per  lb. ;  and  26  lb.  at 
86  cts.  per  lb.,  what  is  one  pound  of  the  mixture  worth? 

Ans.  56  cents. 

6.  If  I  mix  11  lb.  of  tea  at  30  cts.;  30  lb.  at  33  cts.; 
26  lb.  at  67  cts. ;  and  23  lb.  at  86  cts.,  what  is  one  pound 
of  the  mixture  worth?  Ans.  56  cts. 

7.  If  I  mix  30  lb.  of  tea  at  30  cts.;  41  lb.  at  33  cts.; 
23  lb.  at  67  cts. ;  and  49  lb.  at  86  cts.,  what  is  one  pound 
of  the  mixture  worth?  Ans.  56  cts. 


268 


RATIO. 


8.  If  I  mix  41  lb.  of  tea  at  30  cts. ;  30  lb.  at  33  cts.; 
26  lb.  at  67  cts. ;  and  49  lb.  at  86  cts.,  what  is  one  pound 
of  the  mixture  worth  ?  Aiis.  56  cts. 

9.  If  I  mix  41  lb.  of  tea  at  30  cts.;  11  lb.  at  33  cts. ; 
49  lb.  at  67  cts. ;  and  26  lb.  at  86  cts.,  Avhat  is  one  pound 
of  the  mixture  worth  ?  Ans.  56  cts.  • 

10.  If  I  mix  11  lb.  of  tea  at  30  cts.;  41  lb.  at  33  cts.; 
49  lb.  at  67  cts.;  and  23  lb.  at  86  cts.,  what  is  the  mean 
price?  Ans.  56  cts. 

11.  During  14  hours  on  a  certain  day  the  mercury  of 
a  thermometer  stood  as  follows :  2  hours  at  60  degrees ; 
3  hours  at  62  degrees ;  4  hours  at  64  degrees ;  3  hours 
at  67  degrees ;  1  hour  at  72  degrees ;  and  1  hour  at  75 
degrees.  What  was  the  mean  temperature  during  the 
time?  Ans.  65  degrees. 

ALLIGATION    ALTERNATE. 
260.    To  find  the  quantity  that  may  be  used  of  each 
ingredient  of  a  proposed  compound,  the  price  of  each 
being  given,  and  the  mean  price  of  the  compound. 

EXAMPLES. 

1.  A  farmer  has  wheat  worth  $1.10  per  bu. ;  wheat 
worth  60  cts.  per  bu. ;  and  wheat  worth  70  cts.  per  bu. 
He  desires  to  mix  it  so  that  a  bushel  may  be  worth  75 
cents.     What  quantity  of  each  may  be  used  ? 


75 


110 

60 

70- 


OPERATION. 

(1)        (2)       (3)    (4)     (5) 


15 

5 

3 

1 

4 

35, 

7 

7 

35 

7 

7 

Am.  4  bu.  at  $1.10;  7  bu.  at  60  cts.;  and  7  bu.  at  70  cts. 


KATIO.  269 

Explanation. — 110,  which  is  greater  than  the  mean  rate,  is 
connected  by  a  dotted  line  with  60,  which  is  less  than  the  mean 
rate;  also,  110  is  connected  with  70,  forming  another  couplet, 
one  term  of  which  is  greater,  and  the  other  less  than  the  mean 
rate,  75. 

In  column  (1),  15  is  the  difference  between  60  and  75,  and  35  is 
the  difference  between  110  and  75 — each  difference  being  opposite 
the  other  term  of  the  couplet. 

In  column  (2),  5  is  the  difference  between  75  and  70,  and  35  is 
the  difference  between  110  and  75 — each  difference  opposite  the 
other  term  of  the  couplet. 

Column  (3)  is  formed  by  dividing  the  terms  of  (1)  by  their 
greatest  common  divisor.  \     . 

Column  (4)  is  formed  by  dividing  the  terms  of  (2)  b;y  their 
greatest  common  divisor. 

Column  (5)  is  formed  by  adding  the  corresponding  terms  of  (3) 
and  (4). 

Any  number  of  answers  may  now  be  found. 

OPERATION    CONTINUED. 

(6)     (7)     (8)     (9)    (10)  (11)  (12)  (13)  (14)  (15) 


7 

10 

13 

5 

6 

7 

9 

1 

6 

1 

14 

21 

28 

7 

7 

7 

14 

2 

10 

1 

7 

7 

7 

14 

21 

28 

21 

1 

■  5 

4 

Columns  (6),  (7),  and  (8)  are  found  by  multiplying  the  terms 
of  (3)  by  2,  3,  and  4,  respectively,  and  adding  the  corresponding 
terms  of  (4). 

Columns  (9),  (10),  and  (11)  are  found  by  multiplying  the  terms 
of  (4)  by  2,  3,  and  4,  respectively,  and  adding  the  corresponding 
terms  of  (3). 

Column  (12)  is  found  by  multiplying  the  terms  of  (3)  by  2,  and 
those  of  (4)  by  3,  and  then  adding  the  corresponding  products. 

Column  (13)  is  that,  of  (6)  divided  by  7. 

Column  (14)  is  (13)  multiplied  by  5.     (Vide  268,  Ex.  2.) 

Column  (15)  is  (11)  divided  by  7.  All  the  answers  may  be  ver- 
ified by  268,  and  as  many  others  as  one  is  curious  to  find. 

2.  I  wish  to  mix  tea  at  30  cts. ;  tea  at  33  cts. ;  tea  at 
67  cts. ;  and  tea  at  86  cts.  per  pound,  so  as  to  make  the 


270 


llATIO. 


What  quantity  of 


mixture  worth  56  cts.  per  pound, 
each  can  be  used  ? 

OPERATIONS. 


56 


Multiply  or  divide  the  terms  in  columns  (1)  or  (2) 
by  any  numbers  whatever  (merely  preserving  their 
ratio)  for  other  answers.     Thus, 


( 

;i.) 

(2.) 

(1)  (2)  (3)                  (1)  (2)  (3) 

30 30 

30 

30 , 

11 

11 

33  1 

11 

11 

56 

33~4n 

30 

30 

67 •• 

23 

23 

67 -^ 

26 

26 

86 — 26 

26 

86 — ' 

23 

23 

(Vide 

268, 

Ex. 

5.) 

(Vide 

268, 

Ex.  ( 

>.) 

15 

15 

45 

22 

11 

^ 

11 

22 

44 

30 

60 

30 

23 

46 

92 

52 

26 

13 

13 

13 

39 

23 

46 

23 

,30 


56 


130- 

!33=1  I 
|67— li 
186 -^'26 


(3.) 

(1)  (2)  (3)  (4) 


30 


23 


56 


30^ 
33 

67- 


(4.) 

(1)(2)(3 

30 
30 


(Vide  268,  Ex.  7.) 


(Vide  268,  Ex.  8. 


Change  the  numbers  in  columns  (1),  (2),  and  (3)  at 
pleasure,  merely  preserving  their  ratio,  and  add  the  cor- 
responding terms,  for  other  answers. 
(5.) 


(1)  (2)  (3)  (4) 


zw 


30111 


261 


26 


56 


30. 
33 

67 
86 


(6.) 

(1)  (2)  (3)  (4)  (5) 


11 

26 


30 
30 

23126 


(Vide  268,  Ex.  U.; 


RATIO.  271 

Having  arranged  tlie  several  prices  and  the  mean  price 
as  in  the  examples, 

(1.)  Connect  in  couplets  each  price  that  is  less  than  the 
mean  price  with  one  that  is  greater,  hy  lines  which  may 
he  readily  distinguished  from  each  other. 

(2.)  Select  any  couplet,  and  write  the  difference  between 
each  of  its  terms  and  the  mean  price  opposite  the  other 
term,  in  a  column  to  the  right. 

(3.)  Treat  each  couplet  in  the  same  manner,  forming 
as  many  columns  as  there  are  couplets. 

(4.)  Add  together  the  numbers  standing  opposite  each 
'price,  and  place  the  results  in  an  additional  column. 

These  sums  will  be  the  quantities  that  may  be  used 
of  the  price  opposite  which  they  stand. 

Remark  1. — To  find  other  answers,  divide  or  multiply  the  terms 
of  any  column  by  any  number  whatever,  and  then  add  as  in  (4), 
observing  that  it  is  only  necessary  that  the  two  numbers  in  any 
column  preserve  their  ratio. 

Remark  2. — The  corresponding  terms  of  any  two  or  more  col- 
umns, containing  answers,  may  be  added  together  for  other^ 
answers. 

Remark  3. — If  the  prices  are  so  connected  that  each  price  is 
linked  with  only  one  other  price,  only  one  column  to  the  right  is 
necessary.  (See  above,  operations  (1)  and  (2),  where  only  one 
column  was  necessary. 

Remark  4. — When  the  mean  price  is  the  same  as  one  of  the 
given  prices,  the  latter  need  not  be  included  in  the  operation. 

3.  A  merchant  would  mix  wines,  worth  16  shillings, 
18  shillings,  and  22  shillings  per  gallon,  so  that  the 
mixture  may  be  worth  20  shillings  per  gallon.  What 
quantity  of  each  may  be  used  ? 


272 


RATIO. 


OPERATION. 


20 


The  steps  are  precisely  liko 
those  in  Ex.  1. 


(1)   (2)  (3)  (4)  (5) 

16- 

18- 
22- 

Ans.  1  gal.  of  IG;  1  of  18;  and  3  of  22  shillings. 

The  following  are  among  other  answers  easily  obtained : 

(6)     (7)     (8)     (9)    (10)  (11)  (13)     (14)     (15)     (16) 


2 

1 

1 

2 

1 

1 

4 

2 

2 

1 

3 

2 

3 

4 

1 

1 

1 

3 

5 

* 

9 

1 

1 

1 

2 

3 

4 

2 

4 

9 

10 

5 

7 

9 

4 

5 

6 

8 

14 

10 

28 

(15)  is  half  of  (3)  added  to  9  times  (4). 

(16)  is  the  sum  of  (11),  (13),  and  (14).  (See  above, 
Rem.  2.) 

4.  A  merchant  has  wine  at  40  cts. ;  75  cts. ;  60  cts. ; 

and  48  cts.,  and  he  wishes  to  make  a  mixture  of  all 

worth  50  cts.     How  many  gallons  can  he  use  of  each? 

Remark  5. — Since  the  answers  are  limitless  in  number  and  va- 
riety, it  would  be  difficult  to  obtain  those  which  might  be  given. 
Let  the  pupil  verify  those  which  he  may  find  by  268. 

5.  What  quantity  could  be  used  of  each  to  make  the 
mixture  worth  70  cts.  per  gallon  ?  45  cts.  ?  48  ?  60  ? 
(See  Rem.  3.) 

6.  A  merchant  has  three  kinds  of  sugar,  worth  6,  8, 
and  15  cts.  per  pound.  He  wishes  to  make  a  mixture 
worth  11  cts.  a  pound.  What  quantity  of  each  may  be 
used  ? 

7.  What  quantity  of  each  kind  may  he  use  to  make 
the  mixture  worth  12  cts.?  13  cts.?  14  cts.?  10  cts.?' 
Sets.? 

Remark  6. — If  the  quantity  to  be  used  of  a  given  price  is  fixed, 
multiply  the  terms  of  any  answer,  obtained  as  above,  by  the  quo- 
tient of   the  fixed  quantity,  divided  by  the  number  opposite  the 


RATIO.  273 

price  mentioned.  Thus,  if,  in  Ex.  1,  it  is  required  to  use  21  bushels 
at  $1.10,  multiply  the  terms  of  column  (5)  by  2_i__5i^  qj.  column 
(6)  by  V=3,  or(7)by2TV,  etc. 

Remark  7, — If  the  entire  quantity  to  be  used  is  fixed,  apply  the 
principle  of  263  to  the  fixed  quantity,  using  any  answer  as  the  pro- 
portional numbers.     (Vide  Ex.  2,  Operation  (1);  and  263,  Ex.  14.) 

8.  A  grocer  has  currants  at  4  cts. ;  6  cts. ;  9  cts. ;  and 
11  cts. ;  and  he  wishes  to  make  a  mixture  of  240  lb., 
worth  8  cts.  per  pound.  What  quantity  of  each  may 
be  used  ? 

Am,  72;  24;  48;  96,  or  48;  48;  72;  72,  etc. 

POSITION. 

270.  Position  is  the  operation  of  finding  the  true 
answer  to  a  problem  through  the  aid  of  one  or  two  as- 
sumed answers. 

(1.)  Single  Position  assumes  one  ansiver. 
(2.)  Double  Position  assumes  two  answers. 

271.  Single  position  is  applicable  to  those  problems 
in  which,  if  any  number  is  assumed  as  the  answer,  and 
the  steps,  indicated  by  the  problem,  performed  upon  it, 
the  ratio  of  the  result,  and  the  result  in  the  question.,  is 
the  same  as  that  between  the  assumed  ansiver  and  the  true 
answer.     Hence,  the 

RULE. 

(1.)  Assume  any  convenient  number  to  he  the  true  answer, 

(2.)  Perform  the  operations  indicated  hy  the  pi^oblem 
upon  the  assumed  number. 

(3.)  Make  the  result  the  first  term  of  a  proportion,  the 
RESULT  in  the  problem  the  secoimTterm,  and  the  assumed 
answer  the  third  term. 

The  fourth  term,  found  by  258,  will  be  the  true  answer. 


274  RATIO. 

EXAMPLES. 

1.  The  sum  of  |  and  -J-  of  a  certain  number  is  87 J. 
What  is  the  number  ? 

OPERATION. 

Suppose  24       to  be  the  answer. 

1  of  24  is  8 

1  of  24  is  6 

Then  14  : 87J : : 24 

Multiply  couplet  by  2.  (255.)      28  :  175  : :  24 

Divide  couplet  by  7.  (255.)  4  :    25  : :  24 

Divide  antecedents  by  4.'  (256.)     1  :    25  : :  6  :  150  Ans. 

Proof.— I'of  150+i  of  150=87^. 

2.  What  number  is  that  which  being  increased  by  A, 
I,  aiid  i  of  itself,  the  result  will  be  75  ?         Ans.  36. 

Remark. — In  order  to  avoid  fractions,  assume  tlie  least  common 
multiple  of  the  denominators  of  the  fractions,  or  any  multiple 
of  it. 

3.  What  number  is  that,  the  sum  of  |  and  J  of  which 
is  376  ?     Assume  45  or  90.  A71S.  360. 

4.  What  number  is  that  which  being  increased  by  |, 
I,  and  f  of  itself,  the  result  will  be  48i?        Ans.  15. 

5.  The  difference  between  the  fifth  and  tenth  of  a 
certain  number  is  17.     What  is  the  number? 

Ans.  1^0. 

6.  The  difference  between  |  and  |  of  a  certain  num- 
ber is  15.     What  is  the  number?  Ans.  180. 

7.  The  rent  of  a  farm  is  20  per  cent,  greater  this 
year  than  last  year.  This  year  it  is  $1800.  What  was 
the  rent  last  year?  Ans.  $1500. 


RATIO.  275 

^  OPERATION. 

Suppose  $100  to  be  the  rent  last  year. 

20  per  cent.  (J)  of  §100  is     20 

Then  $120  :  $1800  : :  $100  :  $1500 

8.  If  I  of  a  number  be  multiplied  by  7,  and  |  of  the 
number  itself  be  added  to  the  product,  the  result  will  be 
219.     What  is  the  number?  Ans.  45. 

9.  A  gold  watch  is  worth  ten  times  as  much  as  a 
silver  watch,  and  both  together  are  worth  $132.  What 
is  each  watch  worth?  Ans,  263,  Ex.  10. 

10.  The  rent  of  a  farm  is  10  J^  less  this  year  than 
last  year.  This  year  it  is  $1000.  What  was  it  last 
year?  Aiis.  $1111 1. 

11.  The  difference  between  |  and  |  of  a  number  is 
38f.     What  is  the  number?  Ans.  100. 

12.  The  sum  of  |  and  §  of  a  number  is  920.  What 
is  the  number?  Ans.  720. 

13.  One  fifth  of  all  the  sheep  I  have  die  in  1866,  and 
three  fourths  of  the  remainder  in  1867,  when  I  have 
only  20  sheep  left.     What  number  had  I  at  first. 

Ans.  100. 

14.  A  certain  gentleman,  at  the  time  of  his  marriage, 
agreed  to  give  his  wife  |  of  his  estate,  if  at  the  time  of 
his  death  he  left  only  a  daughter,  and  if  he  left  only  a 
son  she  should  have  I  of  his  property ;  but5  as  it  hap- 
pened, he  left  a  son  and  daughter,  in  consequence  of 
which  the  widow  received  in  equity  $2400  less  than  she 
would  have  received  if  there  had  been  only  a  daughter. 
What  would  have  been  the  wife's  dowry  if  he  had  left 
only  a  son?  A71S.  $2100. 


276  KATIO. 

S72.  Double  Position  is  applicable  to  those  prob- 
lems in  which  the  difference  between  the  true  and  first  as- 
sumed number  is  to  the  difference  between  the  true  and 
second  assumed  number  as  the  first  error  is  to  the  second. 

(1.)  If  the  assumed  numbers  are  both  too  large,  or  both 
too  small,  the  errors  arising  from  them  are  said  to  be 

ALIKE. 

(2.)  If  one  of  the  assumed  numbers  happens  to  be  too 
large,  and  the  other  too  small,  the  errors  are  said  to  be 

UNLIKE. 

RULE, 

(1.)  Assume  any  convenient  number,  and  perform  the 
ojjerations  upon  it  indicated  by  the  problem. 

(2.)  Take  the  differ eyice  between  the  result  and  that 
pointed  out  by  the  problem.     This  difference  is  the  first 

ERROR. 

(3.)  Assume  a  second  number,  and  in  like  manner  find 

the  SECOND  ERROR. 

(4.)  If  the  errors,  as  shown  by  the  results,  are  aljke, 
multiply  the  first  error  by  the  second  assumed  num- 
ber, and  the  second  error i»^  the  first  assumed' nui^- 
BER,  and  divide  the  difference  of^  the  products  by  the 
difference  of  the  errors.  -  The  quotient  is  the^true  an- 
swer.    But, 

(5.)  If  the  errors,  as  shown  by  the  results,  are  unlike, 
multijyly  as  before,  and  divide  the  sum  of  the  products  by 
the  SUM  of  the  errors.  The  quotient  will  be  the  true 
answer. 

EXAMPLES. 

1.  A  merchant  expended  ^1500  of  his  capital  for  the 
support  of  his  family  a  year.     At  the  end  of  the  year, 


RATIO. 


277 


however,  lie  has  added  to  the  capital  not  expended  a  sum 
equal  to  3  times  that  part,  thereby  tripling  his  original 
capital.     With  what  sum  did  he  begin  the  year? 


OPEKATION. 

Pirst  assumed  No.                  $2000           $3000 
1500             1500 

(2.) 

Second  assumed  No. 

$500X3= 

$500 
$1500 

$1500 
$4500= 

=$1500X3 

Result  too  small, 
Capital  X3= 

$2000 
$6000 

$6000 
$9000 

Result  too  small. 
Capital  X3. 

$6000-$2000=First 

error.  $4000 

3000 

$3000=Second  error. 
2000 

12000000 
6000000 

60000 

00 

1000)6000000 

$6000    A 

ns. 

See  Rule  (4.) 

(1.) 

First  assumed  No. 

SECOND    OPERATION. 

$2400          $9000 
1500            1500 

(2.) 

Second  assumed  No. 

$900 
$2700 

$7500 
$22500 

Result  too  small, 
Capital  X3= 

$3600 
$7200 

$30000 
$27000 

Result  too  large. 
Capital  X3. 

$7200— $3600=First 

error.  $3600 

9000 

$3000         Second  error. 
2400 

32400000 
7200000 

720000 

0 

6600)39600000 

See  Rule  (5.) 

$6000     Ans. 

278  RATIO. 

2.  Divide  tlic  number  13  into  two  parts,  so  tliat  9 
times  one  of  the  parts  may  be  equal  to  17  times  the 
other. 


OPERATION. 

(1-) 

(2.) 

Assume  10 
9 

and 

3 
17 

Assume     7 
9 

and 

6 

17 

90 
51 

51 

63 

102 
63 

.39 

7 

39 
6 

errors  un 

dike.          39 
3 

39 
10 

273 
390 

234 

117 

234 

390 

78)663  78)351 

8^  one  part.  4i  other  part. 

3.  Divide  100  into  two  parts,  so  that  if  \  of  one  of  the 
parts  be  subtracted  from  \  of  the  other,  the  remainder 
may  be  11.  Ans.  24  and  76. 

4.  A  and  B  had  §85  between  them,  and  f  of  A's 
added  to  f  of  B's  money  make  §60.  How  much  had 
each?  Ans.  A$50;B  §35. 

5.  One  half  a  certain  number  is  the  same  as  one  third 
another  number.  But  if  5  is  added  to  the  first,  and  10 
to  the  second,  then  one  fifth  of  the  first  is  the  same  as 
one  eighth  of  the  second.     What  are  the  numbers? 

Ans.  20  and  30. 

6.  Triple  a  certain  number  is  the  same  as  double  an- 
other number.  But  if  10  is  added  to  the  first,  and  5 
subtracted  from  the  second,  then  5  times  the  sum  will 
be  equal  to  6  times  the  difi'erence.  What  are  the  num- 
bers ?  Ans.  20  and  30. 


■^'^l'^s>\^^j>-4..^. 


281 


7.  In  a  mixture  of  wine  and  Avater,  J  the  whole  +25^ 
gallons  was  wine;  J  the  whole  — 5  gallons  was  water. 
What  was  the  quantity  of  each  in  the  mixture? 

Ans.  Wine  85;  water  35  gallons. 

8.  A  gentleman  supported  himself  3* years  for  50  £  a 
year,  and  at  the  end  of  each  year  added  to  that  part  of 
his  capital  not  thus  expended  a  sum  equal  to  |  of  this 
part.  At  the  end  of  3  years  his  original  capital  was 
doubled.     With  what  sum  did  he  begin  business? 

Ans.  74:0  £. 

""Eemark. — 200 £  and  290 £  are  convenient  suppositions. 

9.  A  boy  had  a  number  of  marbles.  He  laid  aside  2, 
and  then  Avon  in  play  as  many  as  he  had  left.  He  then 
laid  aside  3,  and  Avon  in  play  as  many  as  were  left.  He 
noAv  lays  aside  4,  and,  on  winning  as  many  as  he  had 
left,  finds  he  has  13  in  all.  How  many  did  he  begin 
with?  Ans.  5. 

10.  A  gentleman  has  tAVO  horses,  and  a  saddle  worth 
$50.  If  the  saddle  be  placed  on  the  first  horse  it  makes 
his  value  double  that  of  the  second ;  but  if  the  saddle 
be  placed  on  the  second  horse,  it  makes  his  \'alue  triple 
that  of  the  first. .  What  was  the  value  of  each? 

Ans.  First  $30 ;  second  $40. 

Remark. — If  the  first  horse  is  assumed  to  be  worth  §70,  then  by 
the  frst  condition  of  the  problem  the  second  horse  must  be  worth 
($504-§70)-r-2,  or  $60,  and  the  second  condition  of  the  problem  is 
not  filled  by  $100. 

If  the  first  horse  is  assumed  to  be  worth  $90,  then  the  second 
must  be  worth  ($90-f$50)^2,  or  $70,  and  the  second  error  will 
be  $150. 

11.  Divide  55  into  tAvo  parts,  so  that  the  less  part  di- 
vided by  the  difference  of  the  parts  shall  be  2. 

Ans.  33  and  22. 


278  ,^ 

RATIO. 

12.  A  gentleman  was  asked  the  time  of  day,  and  re- 
plied tliat  I  of  the  time  past  noon  was  equal  to  /g  of 
the  time  to  midnight.     What  was  the  time? 

An8.  12  minutes  past  3  o'clock. 

Remark.— In  tlie  first  place  f  :  -/j  : :  i  :  ^3  : :  11 :  4.  (Vide  252.) 
So  that  the  reply  was  the  same  as  if  he  had  said  that  11  times 
the  number  of  minutes  past  noon  were  equal  to  4  times  the  number 
of  minutes  to  midnight. 

13.  In  a  mixture  of  corn  and  wheat,  \  the  whole  -|-5 
bushels  was  corn,  and  \  the  whole  +10  .bushels  was 
wheat.     What  was  the  quantity  of  each? 

14.  The  sum  of  the  first  and  second  of  three  numbers 
is  13;  of  the  first  and  third  19;  of  the  second  and  third 
24.     What  are  the  numbers?  An%.  4;  9;  and  15. 

15.  A  and  B  have  the  same  income.  A  saves^  J  of 
his  annually;  but  B,  by  spending  |50  per  annum  more 
than  A,  at  the  end  of  6  years  finds  himself  §150  in 
debt.     What  is  the  income  of  each? 

Am.  $125  per  year. 

16.  A  commenced  trade,  and  at  the  end  of  the  third 
year  found  his  original  stock  tripled.  Had  his  gains 
been  $1000  per  year  more  than  they  actually  were,  he 
would  have  doubled  his  stock  each  year.  What  was  his 
original  stock  ?  An%.  |1400. 

17.  Three  persons.  A,  B,  and  C,  were  seen  traveling 
in  the  same  direction.  At  first  A  and  B  were  together, 
and  C  12  miles  in  advance  of  them.  A  goes  7,  B  10, 
and  C  5  miles  per  hour.  In  what  time  will  B  be  half- 
way between  A  and  C?  How  long  before  C  will  be  mid- 
way between  A  and  B?  How  long  since  A  was  midway 
between  B  and  C? 

Ann.  Respectively,  1  h.  30  m.,  3  h.  25 f  m.,  and  12  h. 


INVOLUTION.  281 


INVOLUTION. 


2^72.  (1,)  The  first  poiver  of  a  number  is  the  num- 
be'r  itself. 

(2.)  The  second  power  or  square  of  a  number  is  the 
number  multiplied  by  itself.  Thus,  the  square  of  1  is 
1X1=1;  the  square  of  2  is  2x2^4;  the  square  of  3 
is  3X3=9;  the  square  of  25  is  25x25=625. 

(3.)  The  tliird  power  or  cube  of  a  number  is  the  pro- 
duct of  a  number  taken  as  a  factor  three  times.  Thus, 
the  cube  of  1  is  1X1X1=1;  the  cube  of  2  is  2x2x2 
=8;  the  cube  of  3  is  3x3x3=27;  the  cube  of  4  is 
4X4X4=64. 

(4.)  Any  given  power  of  a  number  is  indicated  by  a 
small  figure  placed  at  the  right  and  a  little  above  the 
number.  Thus,  2*  denotes  that  the  fourth  power  of  2  is 
to  be  taken,  and  that  2  is  to  be  taken  as  a  factor  4  times. 

The  small  figure  is  called  the  exponent  of  the  number 
or  index  of  the  poiver. 

The  expression  2^=16,  is  read  fourth  poiver  of  2 
equals  16. 

5573.  Involution  is  the  operation  of  finding  any 
givcn_pow^j'  Uf  i"  "^^1^^^ er. 

GENERAL    EULE. 

3Iultiply  the  number  by  itself  till  it  is  used  as  a  factor 
as  many  times  as  there  are  units  in  the  exponent. 

Remark. — Fractions  should  always  be  in  their  lowest  terms  be- 
fore multiplying. 

24 


282  INVOLUTION. 

EXAMPLES. 

1.  Find  the  values  of  7^  8^;  25^;  and  125^ 


(1.) 

1st  power,  7 
7 

OPERATIONS 

(2.) 
8 
8 

64 
8 

512 

(3.) 
25 
25 

625 

25 

(4.) 
125 
125 

Square,   49 

7 

15625 
125 

Cube,    343 

15625 

1953125 

2.  Find  the  values  of  11^;   12^;    13^;    lOP;    111  2; 
and  1111^ 

3.  Find  the  values  of  14^;  15';  16^;  17"^;  21*^;  24*; 
and  27^ 

4.  Find  the  values  of  15^;  25^;  35^;  45^;  55'^;  65^; 
75^;  852;  952^ 

Remark. — The  value  of  these  expressions  may  be  found  men- 
tally.    Thus, 

152=10X20+25;  252=20X30+25;  352=30X40+25,  etc. 

5.  Find  the  values  of  15^;  25^;  35^;  45^;  55^;  65^; 
753 ;  85^;  95^ 

6.  Find  the  values  of  2*;  3*;  4*;  5*;  6^;  7^  8^ 

Ans.  In  Table,  page  283. 

7.  Find  the  squares  of  f ;  | ;  £f^ ;  and  y'^g^^-. 

^'^'5.  J;  i;  -,^g;  and  -J^, 

8.  Find  the  squares  of  |;  |J;  ||;  and  ^|. 

Ans.  4;   /g;  ^1;  and  ^4^. 

9.  Find  the  squares  of  2^;  6J  ;  31;  16}/,  and  51. 

A71S.  157,  Ex.  2,  etc. 
10.  Find  the  cube  of  5^  and  the  4th  power  of  S^. 

Ans.  157,  Ex.  28  and  29. 


INV'OLUTIOX. 


283 


11.  Find  the  4tli  power  of  .025. 

Ans.  .000000390625. 

12.  Find  the  5th  power  of  .029. 

Ans.  .000000020511149. 

13.  Find  the    first  nine  powers  of   all  the  numbers 
below  10. 

TABLE. 


02 

o 

o 

n 

I? 
s 

rji 

a 

H 

2 

1 

2 
3 

4 
5 
6 
7 
8 
9 

1 

4 

9 
16 
25 
36 
49 
64 
81 

1 

8 
27 
G4 
125 
216 
343 
512 
729 

1 

16 

81 

256 

625 

1296 

2401 

4096 

6561 

1 

32 

243 

1024 

3125 

7776 
16807 
32768 
59049 

1 

64 

729 

4096 

15625 

46656 

117649 

262144 

531441 

1 

128 

2187 

16384 

78125 

279936 

823543 

2097152 

4782969 

1 

256 

6561 

6553G 

390625 

1679616 

5764801 

16777216 

43046721 

1 

612 

19683 

262144 

1953125 

10077696 

40353607 

134217728 

387420489 

274.    Several  useful  facts  may  be  gathered  from  the 
table. 

(1.)  Any  power  of  1  is  1. 

(2.)  If  the  square  of  a  number  ends  with  1,  the  num- 
ber itself  ends  with  1  or  9. 

(3.)  If  the  square  of  a  number  ends  w^ith  4,  the  num- 
ber itself  ends  with  2  or  8. 

(4.)  If  the  square  of  a  number  ends  with  6,  the  num- 
ber itself  ends  with  4  or  6. 

(5.)  If  the  square  of  a  number  ends  with  9,  the  num- 
ber itself  ends  with  3  or  7. 

(6.)  If  the  cube  of  a  number  ends  with  either 

1;  2;  3;  4;  5.;  6;  7;  8;  9,  the  number  itself 
ends  with    1 ;  8;  7;  4;  5;  6;  3;  2;  9. 


284  INVOLUTION. 

(7.)  The  5th  power,  9th  power,  13th  power,  etc.,  of  a, 
number  ends  Avith  the  same  figure  as  the  number  itself. 

(8.)  It  is  evident  that  if  any  power  of  a  number  ends 
with  a  0,  the  number  itself  ends  with  a  0. 

575.  Any  powder  of  a  number,  multiplied  by  any 
other  power  of  the  same  number,  produces  a  power 
whose  index  is  the  ^  sum  of  the  exponents  of  the  given 
number.  Thus  3"^X3*^3^  and  7^X7'=7^  which  may 
be  verified  by  the  table.     Hence, 

576.  To  find  a  given  power  of  a  number,  when  all 
the  low^er  powers  are  not  wanted, 

3Iultiply  any  tivo  or  more  powers  iogetlier^  the  sum  of 
whose  indices  equals  the  index  of  the  required  p>oicer. 

EXAMPLES. 

1.  Find  the  9th  power  of  3. 

Ans.  3-^X3*-=243x81-=19683  Ans. 

2.  Find  the  15th  power  of  2. 

Ans.  2^X2'--256X  128-32768  Ans. 

3.  Find  the  13th  power  of  3.  Ans.  3^ X  3*^=1594323. 

4.  Find  the  20th  power  of  2. 

Ans.  2^X29X22=1048576-=512X512X4. 
2T7.  Any  number  squared  produces  a  number  con- 
sisting of  exactly  double  the  figures  in  the  number  itself, 
or  of  one  less  Hum  double  the  figures.  For,  1  is  the 
smallest  number  consisting  of  one  figure,  and  9  is  the 
largest;  10  is  tlie  smallest  number  consisting  of  two 
figures,  and  99  is  the  largest;  100  is  the  smallest  of 
tliree  figures,  and  999  the  largest.     Now  we  have 

V^l ;        10---100  ;        100-  -10000  ; 
and         9---81 ;      99^=9801 ;      999^-998001 ;    etc. 


\ 


EVOLUTION.  285 

ST8.  Any  number  cubed  produces  a  number  consist- 
ing of  exactly  triple  the  figures  in  the  number  itself,  or  of 
one  or  two  less  than  triple  the  figures.     For, 

P-=  1 ;  10^=1000 ;  lOO^^-^lOOOOOO  ; 

d'=^n9 ;      99^--970299 ;      999'^=997002999. 


EVOLUTION 


S79.  Any  given  root  of  a  number  is  the  number 
whicli,  on  being  taken  as  a  factor  as  many  times  as  is 
indicated  by  the  name  of  the  root,  will  produce  the 
number  itself.     Thus  : 

(1.)  The^Vs^  root  of  a  number  is  the  number  itself. 

(2.)  Th'e  second  or  square  root  of  a  number  is  one  of 
its  two  equal  factors.  Q^hus  :  the  square  root  of  1  is  1, 
because  1X1=1;  the  square  root  of  4  is  2,  because 
2X2=4;  the  square  root  of  9  is  3,  because  3x3=9. 

(3.)  The  cube  or  third  root  of  a  number  is  one  of  its 
three  equal  factors.  Thus :  the  cube  root  of  1  is  1,  be- 
cause 1X1X1=1;  the  cube  I'oot  of  8  is  2,  because 
2X2X2=8;  the  cube  root  of  27  is  3,  because  3X3X 
3=27;  and  the  cube  root  of  64  is  4,  because  4X4X 
4=64. 

(4.)  The  4th;  5th;  6th,  etc.,  root  of  a  number  is  one 
of  its  4;  5;  6,  etc.,  equal  factors.  Thus:  the  4th  root 
of  1  is  1,  because  1x1X1X1=1 ;  the  4th  root  of  625 
is  5,  because  5x5X5X5=625;  the  5th  root  of  7776  is 
6,  because  6x6x6x6x6=7776. 

(5.)  A  perfect  square  is  a  number  which  has  an  exact 


286  EVOLUTION. 

SQUARE  BOOT.  Tlius,  16 ;  64;  81;  and  144  are  perfect 
squares. 

(6.)  A  perfect  cube  is  a  number  wliicli  has  an  exact 
cube  root.     Thus,  8 ;  27 ;  64,  etc.,  are  perfect  cubes. 

(7.)  Any  given  root  of  a  number  is  indicated  by  a 
fractional  exponent,  or  by  the  aid  of  the  character  ]/, 
called  the  radical  sign.     Thus  : 

The    square    root  of   16  is  indicated  by  16^,  or  by 

l/l6;  the  cube  root  of  8  is  indicated  by  8%  or  by  V'8; 

the  7th  root  of  2187  is  indicated  by  2187^  or  by  f  2187. 
The  figure  at  the  left  of  the  radical  sign  is  called  the 
index  of  the  root. 

(8.)  A  surd  is  a  number  which  requires  a  radical  sign 

or  exponent  to  exactly  express  it.  Thus,  l/2;  1^4;  5^ 
are  surds. 

280.  Evolution  is  the  operation  of  finding  any 
given  root  of  a  number. 

SQUARE  ROOT. 

281.  The  extraction  of  the  square  root  of  a  number  is 
the  operation  of  finding  a  number  which  multiplied  by 
itself  will  produce  the  given  number. 

282.  The  square  root  of  any  integral  number  which 
is  a  perfect  square  and  less  than  400,  may  be  found  from 
memory,  or  by  inspecting  Table  I  of  the  Appendix. 

c 

EXAMPLES. 

1.  What  is  the  square  root  of  1?  4?  9?  16?  25? 
36?     (Vide  273,  Ex.  13.) 


EVOLUTION. 


287 


2.  What  is  the  square  root  of  49?  64?  81?  100? 
121? 

3.  What  is  the  square  root  of  196?  324?  361?  169? 

4.  What  is  the  square  root  of  225?  289?  256?  400? 

IIemark, — All  the  integral  numbers  less  than  400,  other  than 
those  in  the  last  four  examples,  have  only  approximate  square  roots; 
that  is,  their  square  roots  are  surds.  The  same  is  true  of  all 
numbers  not  perfect  squares. 

S83.  The  general  method  of  extracting  the  square 
root  of  a  number  may  be  readily  understood  from  the 
following  operations.  We  will  first  square  some  num- 
ber, as  37. 

OPERATION   FIRST. 

(1.)  (2.) 

37  30+7 

37  30+7 


49 
210 
210 
900 

1369 


30x7+7^ 
302+30X7 

30^+2x30x7+7^ 


In  (2)  it  is  evident  that  30x7  added  to  30X7  is  twice 
30X7,  or  2X30X7.  To  reverse  these  steps,  that  is,  to 
extract  the  square  root  of  1369,  we  proceed  as  follows: 


OPERATION   SECOND. 


1369 1 37              30 
9      ^ 

469          2x30+7 

469 



30^+2x30x7+7^  30+7 
302                           

2x30X7+7' 
2x30x7+7' 

67 


Explanation, — By  putting  a  point  or  dot  over  every  alternate 
figure   of   the    given    number,   commencing   with    the    right-hand 


288  EVOLUTION. 

figure,  we  not  only  can  determine  the  number  of  figures  of  wliicli 
the  root  will  consist,  (vide  277,)  but  we  can  also  determine  the 
left-hand  figure  of  the  root;  for  the  square  root  of  the  largest  per- 
fect square,  less  than  13,  must  be  that  figure.  Now  9  is  the  largest 
perfect  square  less  than  13,  and  its  square  root  is  3.  This  3  is 
written  at  the  right  and  also  at  the  left  of  the  given  number. 

Having  taken  9  from  13,  the  remainder,  469,  is  the  same  as  2X 
30X7-j-7^.  Now  if  we  double  the  3  just  obtained,  it  in  eifect  pro- 
duces 60,  which  is  2X30.  If  we  now  divide  469  by  60,  and  annex 
the  quotient  to  6,  and  also  to. the  right  of  the  root,  as  Operation  (2) 
points  out,  and  then  multiply  67  by  7,  the  exact  remainder,  469, 
will  be  produced  whenever  the  given  number  is  a  perfect  square. 
Hence, 

284.  To  extract  the  square  root  of  any  integral 
number, 

(1.)  Divide  the  number  into  periods  of  ttvo  figures  each, 
counting  from  right  to  left. 

(2.)  Fbid  the  greatest  perfect  square  in  tJie  left-hand 
period,  and  'place  its  square  root  at  the  left  and  also  at 
the  right  of  the  given  number.  Subtract  this  greatest 
perfect  square  from  the  left-hand  p)eriod,  and  to  the  re- 
mainder annex  the  next  period  to  the  right;  the  result  is 
the  First  Dividend. 

(3.)  Double  the  figure  of  the  root  just  found  and  place 
it  to  the  left  of  the  first  dividend.  See  how  many  times 
tills  double  figure,  with,  a  cipher  mentally  annexed,  is  con- 
tained in  the  first  dividend.  Place  the  quotient  figure  as 
the  second  figure  of  the  root,  and  also  annex  it  to  double 
tJt^  first  figure  as  the  First  Divisor. 

(4.)  Multiply  the  first  divisor  by  the  second  figure  of 
the  root,  and  subtract  the  ijroduct  from  the  first  dividend, 
and  to  the  remainder  annex  the  next  period  to  the  right 
for  the  Second  Dividend. 


EVOLUTION. 


289 


(5.)  Double  iJie  part  of  the  root  now  found  and  place 
it  to  the  left  of  the  second  dividend ;  see  how  many  ti7nes 
this  double  root,  luith  a  cipher  mentally  annexed,  is  con- 
tained in  the  second  dividend,  and  place  the  quotient  at 
the  third  figure  of  the  root,  and  also  to  the  right  of  double 
the  previous  pa7i  of  the  root,  as  the  Second  Divisor. 

Proceed  with  the  second  divisor  as  with  the  first,  and 
continue  the  operation  till  all  the  periods  have  been  used. 

Remark  1. — Double  the  root  already  found  is  sometimes  called 
the  trial  divisor. 

Remark  2. — It  sometimes  happens,  as  in  Operation  (2)  below, 
that  in  making  the  division  for  the  next  figure  of  the  root  the 
quotient  is  too  large.  In  such  a  case  make  the  quotient  figure  so 
small  that  the  product  will  be  less  than  the  dividend. 

Remark  3. — It  sometimes  happens,  as  in  Ex.  15,  Operations  (1) 
and  (2)  below,  that  a  dividend  is  too  small  to  contain  the  trial 
divisor.  In  such  a  case  place  a  cipher  in  the  root  and  to  the  right 
of  the  trial  divisor,  and  then  annex  the  7iezt  period  to  the  dividend. 

Remark  4. — In  the  case  of  decimals,  the  first  two  figures  on  the 
right  of  the  decimal  point  constitute  one  period,  the  next  two 
figures  another,  and  so  on. 

EXAMPLES. 

1.  Find  the  square  roots  of  6084;  1521;  and  98.01. 


148 


6084178  Ans. 
49 


1184 
1184 


OPERATIONS. 

(2.) 

152i|39  Ans. 
9 


69 


621 
621 


18.9 


(3.) 

98.0i|9.9  Ans. 
81 


17.01 
17.01 


2.  Find  the  square  roots  of  2025;  3969;  6241;  and 
6561.  Ans.  45;  63;  79;  and  81. 

25 


290  EVOLUTION. 

3.  Find  the  square  roots  of  1936;  2401;  8281;  and 
8836.  Ans.  44;  49;  91;  and  94. 

4.  Find  the  square  roots  of  10.24;    10.89;    11.56; 
12.25.  Ans.  3.2  ;  3.3  ;  3.4  ;  3.5. 

5.  Find  the  square  roots  of  6400 ;  2500 ;  841 ;  and 
729.  Ans.  80 ;  50;  29;  and  27. 

6.  What  is  the  value  of  i/8281+t/6561+i/50.41. 

Ans.  179.1. 

7.  What  is  the  value  of  1/10:89+^1444+1/18.49. 

Ans.  45.6. 


8.  What  is  the  value  of  t/5776— 1/40.96. 

Ans.  69.6. 

9.  What  is  the  value  of  l/5625— 1/3025. 

Ans.  20. 


10.  What  is  the  value  of  V  9025— l/l225. 

Ans.-60. 

11.  What  are  the  square  roots  of  21904  and  3564.09. 

OPERATIONS. 


1 

21904  1 
1 

.48  Ans. 
the  square 

5 

109 

118.7 

(2.) 

3564.09   59.7  ^ws. 
25 

24 

119 
96 

1064 
981 

288 

2304 
2304 

83.09 
83.09 

12.  What  are 

roots  of 

332929  and  467856. 
ins.  577  and  684. 

13.  Find  the  value  of  t/473344+i/48.8601. 

Ans.  694.99. 


EVOLUTION.  291 


14.  Find  the  value  of  1-/44.0896+1/3080.25. 

Ans.  62.14. 

15.  Find  the  square  roots  of  366025  and  49126081. 


OPERATIONS. 

(1.)  (2.) 


6 
1205 


366025  I  605  Ans.  7 

36    


6025         14009 
6025 


49126081 1  7009  Ans. 
49      


126081 
126081 


16.  Find  the  square  roots  of  259081  and  826281. 

A71S.  509  and  909. 

17.  Find  the  square  roots  of  49.4209 ;  404.01 ;  and 
.822649.  Ans.  7.03  ;  20.1 ;  and  .907. 

18.  Find  the  square  roots  of  12321 ;  1234321 ;  and 
123454321. 

19.  Find  the  square  roots  of  49284  and  4937284. 

20.  What  is  the  value  of  l/ 110889— l/ 40376081. 

Ans.  312.91. 

21.  Find  the  square  root  of  .0011943936. 

Ans.  .03456. 

22.  Find  the  square  roots  of  99980001  and  9999- 
800001. 

23.  Find  the  square  root  of  152399025. 

Ans.  12345! 

24.  Find  the  square  root  of  2950771041. 

Ans.  54321. 

25.  Find  the  square  root  of  8264446281. 

26.  Find  the  square  root  of  6529932864. 

27.  Find  the  square  root  of  4999479849. 


292 


EVOLUTION. 


28.  Find  the  square  root  of  2,  and  also  of  3,  to  7 
decimal  places. 

OPERATIONS. 

(1-)  (2.) 


1 

2.00000011.4142136 

1      

:  1 3.000000|1.7320508 

2.4 

1.00 
96 

400 

281 

2.7 

3.43 

3.462 

2.00 

1.89 

2.81 

1100 
1029 

2824 

11900 
11296 

7100 
6924 

2.8282 

)   60400 
56564 

3.4640 

)   17600 
17320 

3836 
2828 

280 

272 

1008 
848 

160 
168 

Remark. — After  finding  four  figures  of  the  root  as  usual,  the  re- 
maining figures  are  found  by  simply  dividing  the  last  dividend  by 
the  last  divisor,  except  that,  instead  of  annexing  ciphers  to  the 
dividend,  we  take  away  a  figure  at  the  right  of  the  divisor  at  each 
new  figure  of  the  root.  Attention  is  of  course  paid  to  the  rules  for 
division  of  decimals.     In  general, 

Having  found  one  more  than  half  the  required  figures  of  the  root  in 
the  ordinary  way,  the  remaining  figures  may  he  found  by  dividing  the  last 
dividend  hy  the  last  divisor,  carefully  observing  the  rules  in  division  of 
decimals. 

In  this  manner  Table  II  of  the  Appendix  may  be  verified. 


EVOLUTION.  293 

285.  The  square  root  of  the  product  of  two  numbers 
is  the  same  as  the  product  of  the  square  roots  of  the 
numbers.     Thus : 


l/4x9  is  the  same  as  l/4  Xv'9 ;  also  V2  xS=V2  xVB, 
as  may  be  easily  verified. 

S8G.  To  find  the  square  root  of  a  composite  number, 
Find  the  product  of  the  roots  of  its  factors,  mahing 
one  of  the  factors  a  perfect  square  if  possible. 

EXAMPLES. 

'1.  What  is  the  square  root  of  8? 
■■    We  have  t/8=  1/4x^  =  1/4  X"l/2  =  2Xl/2. 
'     N'ow  t/2=i1.4142136.     Hence,  l/8=2X1.4142136. 

Ans.  2.8284272. 

2.  Find  the  values  of  l^FS;  VM;  VM',  and  1/72. 

Ans.  4.2426408,  etc. 

3.  Find  the  values  of  l/'98 ;  l/l28 ;  VlQ2 ;  and  1/200. 

Ans.  9.899495,  etc. 

4.  Find  the  values  of  Vl2',  VVJ;  t/48;  and  VJb. 

Ans.  i/4x3=2Xt/3=3.464102. 

5.  Find  the  values  of  1/IO8;  l/l47;  T^IM;  and 
1/243.  Ans.  10.392305,  etc. 

6.  Find  the  values  of  l/20 ;  1^28;  1^99;  and  1/8O. 

t/20=2Xi/5;  t/28=2X>^7;  T/99=3X"/li;  VW):^4:XV^. 

7.  Find  the  values  of  1/120  ;  l/270 ;  1/8OO ;  and 
1/450.  Ans.  2X1/30=10.954451,  etc. 

Remark.— In  practice  reject  as  many  figures  on  the  right  of  the 
product  of  the  tabulated  decimal  as  are  found  in  the  root  of  the 
factor  which  is  a  perfect  square. 


294  EVOLUTION. 

287.  The  square  root  of  a  fraction  is  the  square  root 
of  the  numerator  divided  bj  the  square  root  of  the  de- 
nominator.    Thus  : 

The  square  root  of  |  is  -7^  =  1,  because  |X|=|. 

288.  To  find  the  square  root  of  a  fraction, 

(1.)  If  necessary,  multiply  both  terms  of  the  fraction 
hy  the  smallest  number  that  will  make  the  denominator  a 
perfect  square. 

(2.)  Divide  the  square  root  of  the  resulting  numerator 
by  that  of  the  denominator^  for  the  required  root. 

EXAMPLES.  _^ 

1.  What  is  the  square  root  of  J  ?  ^  /  '  ^ 

„,    ,                          ^            /_           ._      1/2       1/2      1.4142L36 
We  have  J=f ;  then  V  |  =  l/f  -:^  — ^= -— 

=.7071068. 

2.  What  are  the  square  roots  of  J;  J  ;  4;  ^^1 

-^Tis.  1x1/3^.5773502. 

3.  What  are  the  square  roots  of  f  ;  | ;  f ;  i  ?     (Vide 

286,  Ex.  6.  Ans.  ^^^^  =.8944272,  etc. 

4.  What  are  the  square  roots  of  f ;  | ;  4  >  I  '•  j 

^Tis.  ]^=.9258201,  etc. 

5.  What  are  the  square  roots  of  2| ;  3 J ;  4§ ;  2  J  ? 

^ns.  2^=1.5811388^  etc. 

6.  What  are  the  square  roots  of  30/(j;  11/^;  Hi?; 
and  6|f  ?  Jtzs.  5';  sf;  3|.;  and  2|. 

7.  What  are  the  square  roots  of  2| ;  18  J^;  272^; 
51  A?  ^ns.li;  41;  5^;  74. 


EVOLUTION.  205 

CUBE    II  GOT. 

289.  The  extraction  of  the  cube  root  of  a  number  is 
the  operation  of  finding  one  of  itfe  three  equal  factors. 

290.  The  cube  root  of  any  integral  number  which  is 
a  perfect  cube  and  less  tlian  1000,  may  be  found  from 
memory,  or  by  inspecting  the  third  column  of  the  Table 
under  273,  Ex.  13. 

EI^AMPLES. 

1.  What  are  the  cube  roots  of  1 ;  8;  27;  64;  12^? 

2.  What  are  the  cube  roots  of  216 ;  343 ;  512 ;  and 

729?  •^ 

Remark. — All  the  integral  numbers  less  than  1000,  other  than 
those  in  the  above  two  examples,  have  only  approximate  cube  roots, 
that  is,  their  cube  roots  are  surds.  (Vide  279,  (8.)  The  same  is 
true  of  all  numbers  not  perfect  cubes. 

291.  The  general  method  of  extracting  the  cube  root 
of  a  number  may  be  drawn  from  the  following  opera- 
tions. We  will  cube  the  number  37  by  continuing 
Operation  (2)  of  283. 


OPERATION   FIRST. 

37^= 

1369= 

30^+2x30x7+72 

37- 

.37= 

30+7 

30^+2X30^x7+30x7^ 

302x7+2x30x7-f7^ 

37^=        50653=  303+3x30^X7+3x30X7^+7^ 

In  the  operation,  the  quantity  30^+2X30X7+7^  is 
first  multiplied  by  30  and  then  by  7,  and  the  two  indi- 
cated products  are  added  together.     It  is  evident  that 


296 


EVOLUTION. 


twice  30^X7  added  to  once  30^X7  is  the  same  as  three 
times  30^X7,  that  is,  the  sum  is  3x30'X7;  also,  that 
once  30X7^  added  to  twice  2>^Y.1^  is  three  times  30 X7^ 
that  is,  the  sum  is  3X30x7^. 
We  will  now  reverse  these  steps. 


1st  Col. 
30 

2X30 
3X30+7 


OPERATION   SECOND. 

2d  Col. 
302 

3X30= 
3X30^-1-3X30X7+72 


Quantity.      Root.  [30+7 

30^+3X302X7+3X30X72+78 
30^ 

3X302X7+3X30X72+73 
3X302x7+3X30X72+73 


The  above  operation  contracted  is  as  follows : 


OPERATION    THIRD. 

Number,  Root, 

50653  I  37. 

27 


1st  Col. 

2d  Col. 

3 

6 
97 

9 

2700 

3379 

. 

23653 
23653 


The  steps  in  Operation  Second  are  as  folloAvs: 
Having  arranged  the  quantity  so  as  to  form  two 
columns  on  the  left,  with  30  in  the  1st  col.,  its  square 
in  the  2d  col.,  and  its  cube  under  the  quantity  itself. 
We  now  indicate  double  the  30  of  the  1st  col.,  and  write 
it  under  30  of  the  same  column.  Next  multiply  this 
2X30  by  30,  which  gives  2X30^,  and  add  the  product  to 
30^  of  the  second  column,  and  write  the  indicated  sum 
or  3x30^  under  30^  in  the  same  column.  Next  indicate 
the  triple  of  the  30  in  the  1st  col.,  and  write  the  indi- 


EVOLUTION.  297 

oated  product,  viz.,  3x30,  under  2X30  of  the  1st  col. 
If  we  now  divide  the  first  term  of  the  remainder,  viz., 
3X30^X7,  by  3x30-,  the  second  term  of  the  2d  col., 
the  quotient  will  of  course  be  7,  the  second  term  of  the 
root.  This  7  is  added  to  the  right  of  3x30  of  the  1st 
col.,  making  3x30+7,  which  is  then  multiplied  by  7, 
and  added  to  3x30^  of  the  2d  col.,  giving  3x30^+3 
X30x7+7^  which  is  now  multiplied  by  7,  and  the  pro- 
duct is  the  same  as  the  remainder,  after  subtracting  30'^ 
from  the  quantity  itself. 

The  indicated  products  and  additions  of  Operation 
Second  are  actually  made  in  Operation  Third,  omitting 
ciphers,  which  would  be  of  no  service  if  written  down. 
Hence, 

!S9S.  To  extract  the  cube  root  of  any  integral 
number, 

(1.)  Divide  the  number  into  periods  of  three  figures 
each,  counting  from  right  to  left.  The  left-hand  period 
may  consist  of  one,  two,  or  three  figures.     (Vide  278.) 

(2.)  Find  bi/  290  the  largest  perfect  cube  less  than  the 
left-hand  period ;  place  its  cube  root  to  the  right  of  the 
number  J  and  also  to  the  left,  as  the  first  term  of  the  1st 
col. ;  square  the  first  term  of  the  1st  col.,  and  write  it  as 
the  first  term  of  the  2d  col.;  cube  the  first  term  of  the 
1st  col.,  and  subtract  it  from  the  left-hand  period  of  the 
number,  and  annex  the  next  period  to  the  right.  The 
result  is  the  first  dividend.  Add  the  figure  in  the  1st 
col.  to  itself,  and  lurite  the  sum  as  the  second  term  of  the 
1st  col.;  midtiply  the  second  term  of  the  1st  col.  by  the 
figure  in  the  root,  and  add  the  product  to  the  first  term 
of  the  2d  col.;  the  sum  with  two  ciphers  annexed  is  the 


298  EVOLUTION. 

second  term  of  the  2d  col.  Add  the  figure  in  the  root 
to  the  second  terin  of  the  1st  col.,  and  place  the  sum  under 
the  second  term. 

(3.)  See  hoiv  many  times  the  second  term  of  the  2d 
col.  is  contained  in  the  fikst  dividend,  and  write  the 
quotient  as  the  second  figure  of  the  root,  and  also  to  the 
right  of  the  last  number  in  the  1st  col.,  forming  its  third 
term;  multiply  the  third  term  of  the  1st  col.  by  this 
second  figure  of  the  root,  and  add  the  product  to  the  second 
term  of  the  2d  col.  as  its  third  ter7n;  multiply  this  third 
term  by  the  second  figure  of  the  root,  and  subtract  the  p)ro- 
ducf  from  the  first  dividend,  and  annex  the  next  period  of 
the  number  for  the  second  dividend. 

Proceed  with  the  second  figure  of  the  root  precisely 
as  with  the  first,  and  continue  the  operation  till  all  the 
periods  have  been  used. 

Remark  1. — It  often  happens  that  in  making  the  division  for 
the  figure  of  a  root  the  quotient  will  be  found  too  large.  In  such 
a  case  erase  the  work  as  far  back  as  that  point,  and  make  the  root 
figure  so  small  that  the  subtrahend  will  be  less  than  the  dividend. 
(Vide  Ex.  13.) 

Remauk  2. — If  a  dividend  is  too  small  to  contain  the  divisor, 
place  a  cipher  in  the  root,  and  also  to  the  right  of  the  last  term 
of  the  first  column,  and  two  ciphers  to  the  last  term  of  the  second 
column.     (Vide  Ex.  19.) 

Remark  3. — In  the  case  of  decimals,  the  first  three  figures  on  the 
right  of  the  decimal  point  constitute  one  period,  the  next  three 
figures  another,  and  so  on. 

Remark  4. — This  method  of  taking  the  cube  root  is  so  superior, 
it  is  not  a  little  surprising  that  it  has  not  been  adopted,  to  the  ex- 
clusion of  all  others.  Aside  from  its  value  in  this  connection,  it 
prepares  the  student  for  an  easy  understanding  of  the  best  mode 
of  finding  the  numerical  values  of  the  roots  of  equations  above 
the  second  degree  in  Algebra. 


EVOLUTIOX. 


299 


EXAMPLES. 

1.  Find  the  cube  root  of  753571. 


OPERATION. 


9 

18 

271 


81 

24300 

24571 


753571  I  91  Ans. 

729 


24571 
24571 


2.  Find  the  cube  roots  of  531441  and  357911. 

Ans.  81  and  71. 

3.  Find  the  cube  roots  of  132651  and  72507. 

A71S.  ^r^  43. 

4.  Find  the  cube  roots  of  970299  and  681472. 

Ans.  99  and  88. 

5.  Find  the  cube  roots  of  456533  and  287496. 

Ans.  77  and  66. 

6.  Find  the  cube  roots  of  6.859  and  24.389. 

Ans.  1.9  and  2.9. 

7.  Find  the  value  of  1^704969+1^421875. 

Ans.  164. 

8.  Find  the  value  of  1^884736+1^314432. 

Ans.  164. 

9.  Find  the  value  of  1^110592—1^^103823. 

Ans.  1. 

10.  Find  the  value  of  f" 91125 Xl^4287o. 

An.s.  1575. 

11.  Find  the  value  of  1^456533—1^421875. 

Ans.  2. 

12.  Find  the  value  of  1^4096X1^15625.   Am.  400. 


300  EVOLUTION. 

13.  Find  the  cube  root  of  2863288. 


OPERATION. 


1 

2 

34 
38 
422 


1 

300 
436 
58800 
59644 


2863288  [142 
1 


1863 
1744 

119288 
119288 


14.  Find  the  cube  roots  of  2299968  and  2352637. 

-^  Am.  132  and  133. 

15.  Find  the  cube  roots  of  10793861  and  14526784. 

Am.  221  and  244. 

16.  Find  the  cube  roots  of  36926037  and  63521199. 

Am.  333  and  399. 

17.  Find  the  cube  root  of  212776173.  Am.  597. 

18.  Find  the  cube  root  of  997002999.  Am.  999. 

19.  Find  the  cube  root  of  743677416. 


OPERATION. 


9 

18 
2706 


81 

2430000 
2446236 


743677416  I  906 

729      

14677416 
14677416 


20.  What  is  the  cube  root  of  ip30301?    Am.  101. 

21.  What  is  the  cube  root  of  128787625? 

22.  What  is  the  cube  root  of  225866,529  ? 

Am.  609. 


EVOLUTION. 


101 


23.  Find   the    cube   root   of   2    to   seven  places   of 
decimals. 

OPERATION. 

2|1.2599211  An90:, 


1 

1 

2 

300 

32 

364 

34 

43200 

365 

45025 

370 

4687500 

3759 

4721331 

3768 

475524300 

37779 

4758)64311 

1.000 

.728 


272000 
225125 


46875000 
42491979 


4383021000 

4282778799 

10024)2201 
9517 


507 
475 

32 
47 


We  proceed  in  the  usual  way,  and  find  1.2599,  at 
which  point  divide  the  dividend  10024  by  4758,  reject- 
ing other  figures  on  the  right  as  not  afi'ecting  the  result. 
In  general,  having  found  one  more  than  half  the  figures 
of  the  decimal  in  the  ordinary  way,  divide  the  last  divi- 
dend hy  the  last  term  of  the  2d  col.,  rejecting  from  the 
right  of  the  dividend  one  figure  less  than  from  the  right 
of  the  divisor.  In  this  manner  Table  III  of  the  Ap- 
pendix may  be  verified. 

24.  Find  the  cube  roots  of  3,  4,  etc.,  to  20. 


302  EVOLUTION. 

29S.  The  cube  root  of  the  product  of  two  numbers 
is  the  same  as  the  product  of  the  cube  roots  of  the  num- 
bers.    Thus  : 

1^8x27-1^8  X1^27,  and  f'2xE=f2xf'S, 
as  may  be  verified  from  the  Table.     Hence, 

294.  To  find  the  cube  root  of  a  composite  number, 
Find  the  product  of  the  cube  roots  of  its  factors,  makiiig 

one  of  the  factors  a  perfect  cube,  if  possible. 

EXAMPLES. 

1.  What  is  the  cube  root  of  16?  We  have  #"^16= 
#"8X2=1^8  X#'2  =2X1^2.  Now,  if  2 -=1.2599211. 
Hence,  1^16=2X1.259921.  Ans.  2.519842. 

2.  Find  the  values  of  #"24^  1^321  1^40^  and  1^48. 
fSxf'^;  1^8X1^4;  1^8X1^5;  1^8X1^6. 

Ans.  2.8845,  etc. 

3.  Find  the  values  of  1^54;  1^128;  #'250. 

Ans.  #^27  X^'g -3.779763. 

4.  Find  the  values  of  ^88^  ^297";  and  ^704. 
(Vide  286,  Rem.)  Ans.  2X^11=4.44796. 

295.  The  cube  root  of  q>  fraction  is  the  cube  root  of 
the  7iumerator  divided  by  the  cube  root  of  the  denomi- 

nator.     Thus,  the  cube  root  of  /^  is  -— ^  =  |,  because 

F  27 

fX|X|=3V     Hence, 

296.  To  find  the  cube  root  of  a  fraction, 

(1.)  If  necessary,  multiply  both  terms  of  the  fraction 
by  the  smallest  number  that  ivill  make  the  denominator  a, 
perfffct  cube. 


EVOLUTION.  303 

(2.)  Divide  the  cube  root  of  the  resulting  numerator  hy 
that  of  the  denominator  for  the  required  root. 

EXAMPLES. 

1.  What  is  the  cube  root  of  i?     We  have 

1=^       Then  irT_iri=^4_f4  _]L587401_ 

.7937005  Ans. 

2.  What  are  the  cube  roots  of  | ;  J ;  J ;  and  I  ? 

Ans.  1X1^9  =.693361,  etc. 

3.  What  are  the  cube  roots  of   f ;    |;    |;    and  li? 

Jxrg^  Jxiri2i  fxif 9]  ixif To: 

^ns.  .908561,  etc. 

4.  What  are  the  cube  roots  of  | ;  | ;  y^g ;  | ;  -J|  ? 

^?is. -I  X  1^18  =.87358. 

5.  What  are  the  cube  roots  of  ff ;  iff;  ||f  ' 

Remark. — When  the  root  of  a  perfect  square,  cube,  fourth  power, 
etc.,  contains  no  more  than  three  figures,  it  may  be  taken  mentally, 
after  a  little  practice,  and  on  observance  of  the  points  in  sec.  274. 
All  roots  of  integral  numbers  containing  not  more  than  six  figures, 
decimals  and  fractions,  are  easily  found  by  a  common  Table  of 
Logarithms.     (Algebra,  152.) 

PROBIiEMS. 

29T.  The  sum  of  two  numbers  and  their  product 
being  given,  to  find  the  numbers. 

Find  the  square  root  of  four  times  the  product 
SUBTRACTED  FROM  the  Square  of  the  sum.  This  root 
will  be  the  diiference  of  the  numbers.  (Vide  125, 
Ex.  10.) 


V 


304  EVOLUTION. 

EXAMPLES. 

1.  The  sum  of  two  numbers  is  105  and  their  product 
2666.     What  are  the  numbers?     l/l052— 4X2666=19. 

Ans.  62  and  43. 

2.  The  sum  of  two  numbers  is  10  and  their  product 
24.  What  are  the  numbers?  Ans.  6  and  4. 

3.  The  sum  of  two  numbers  is  8j  and  their  product 
17 J.     What  are  the  numbers?  A71S.  5  and  3|. 

4.  The  sum  of  two  numbers  is  28  and  their  product 
196.     Wliat  are  the  numbers?  Ans.  14  and  14. 

S98.  The  difference  of  two  numbers  and  their  pro- 
duct being  given,  to  find  the  numbers, 

JFind  the  square  root  of  four  times  the  product  added 
TO  the  square  of  the  difference.     This  root  is  their  sum. 

EXAMPLES. 

1.  The  difference  of  two  numbers  is  19  and  their  pro- 
duct 2666.     What  are  the  numbers  ? 


y  192+4x2666=105.  Ans.  62  and  43. 

2.  The  difference  of  two  numbers  is  3  and  their  pro- 
duct 40.     What  are  the  numbers?  Ans.  8  and  5. 

3.  The  difference  of  two  numbers  is  7  and  their  pro- 
duct 294.     What  are  the  numbers  ?     Ans.  21  and  14. 

4.  The  difference  of  two  numbers  is  3^  and  their  pro- 
duct 73 J.     What  are  the  numbers?     Ans.  10 J  and  7. 

5.  The  sum  of  two  numbers  is  23j  and  their  product 
135.  What  are  the  numbers?  Ans.  11}  and  12. 

6.  The  difference  of  two  numbers  is  J  and  their  pro- 
duct 135.     What  are  the  numbers  ? 


ARITHMIiTICAL    PROGRESSION.  305 


ARITHMETICAL  PROGRESSION. 


299.  An  Arithmetical  Progression  is  a  series  of 
numbers  in  which  any  term  is  found  by  adding  a  given 
number  to  the  preceding  term,  or  by  subtracting  a  given 
number  from  tJie  precediyig  term. 

(1.)  The  common  difference  is  the  number  to  be  added 
or  subtracted. 

(2.)  The  progression  is  increasing  when  the  common 
difference  is  added. 

(3.)  The  progression  is  decreasing  when  the  common 
difference  is  subtracted. 

300.  In  every  progression  these  five  points  may  be 
considered,  viz.,  the  first  term,  the  last  term,  the  common 
difference,  the  number  of  terms,  and  the  sum  of  all  the 
terms.     Thus,  in  the  arithmetical  progression, 

1,  3,  5,  7,  9,  11,  13, 
the  first  t'erm  is  1,  the  last  term  13,  the  common  differ- 
ence 2,  the  number  of  terms  7,  the  sum  of  all  the  terms 
49,  and  the  progression  is  increasing. 
In  the  decreasing  progression, 

22,  19,  16,  13,  10,  7,  4,  1, 
the  first  term  is  22,  the  last  term  1,  the  common  differ- 
ence 3,  the  number  of  terms  8,  and  the  sum  of  all  the 
terms  92. 

(1.)  The  first  and  last  terms  are  sometimes  called 
extremes,  and  the  intermediate  terms  the  means.  The 
means  must  of  course  be  less  by  2  than  the  number  of 
terms. 

20 


306  AEITHMETICAL    PROGRESSION. 

(2.)  In  a  progression  of  three  terms,  the  middle  term 
is  called  the  arithmetical  mean,  and  is  half  the  extremes. 

301.  To  find  the  last  term,  when  the  first  term,  7mm- 
her  of  terms,  and  common  difference  are  known. 

EXAMPLES. 

1.  The  first  term  of  a  progression  is  4,  common  dif- 
ference 3,  number  of  terms  50.     What  is  the  last  term? 

ANALYSIS. 

If  we  take  a  few  terms  of  the  progression ;  thus, 
1st.        2d.  3d.  4th.  5th. 

4,     4+3,     4+2x3,     4+3x3,     4+4x3; 
that  is,  4         7  10  13  16; 

we  see  that  any  term,  as,  for  instance,  the  5th,  is  found 
by  adding  to  the  1st  the  product  of  the  common  differ- 
ence into  a  number  less  by  1  than  that  denoting  the 
term.  If  we  should  continue  the  progression  to  the 
50th  term,  the  quantity  standing  under  50th  Avould 
evidently  be  4+49x3,  and  the  50th  term  would  there- 
fore be  151.     Hence, 

When  the  progres'sion  is  increasing, 

(1.)  To  the  first  term  add  the  product  obtained  by  mul- 
tiplying the  common  difference  by  the  number  of  terms 
less  1. 

2.  In  the  progression  2,  4,  6,  8,  etc.,  what  is  the  20th 
term?  40th  term?  60th?  71st?  100th? 

Ans.  40,  80,  etc. 

3.  In  the  progression  3,  11,  19^  27,  etc.,  what  is  the 
50th  term?  80th?  150th?  200th?  500th? 

Ans.  395,  635,  etc. 


ARITHMETICAL    PROGRESSIOX.  307 

4.  In  the  progression  7,  11,  15,  etc.,  what  is  the  10th 
term?  45th?  75th?  101st?  Ans.  43,  183,  etc. 

5.  In  the  progression  8,  11,  14,  etc.,  what  is  the  12th 
term?  24th?  47th?  81st?  1000th? 

Last  Ans.  3005. 

6.  In  the  progression  1,  1^,  2,  2|,  etc.,  what  is  the 
900th  term?  1200th  term?  1500th  term? 

Ans.  450 1. 

7.  In  the  progression  5,  5-|,  5f ,  6,  etc.,  w4iat  is  the 
45th  term?  90th?  750th?  100th?  Ans.  19f,  etc. 

When  the  progression  is  decreasing, 

(2.)  From  the  first  term  subtract  the  product  obtained 
by  7nultiplying  the  common  difference  by  the  number  of 
terms  less  1. 

8.  In  the  progression  500,  496,  492,  etc.,  what  is  the 
20th  term?  25th?  30th?  50th?  ^^is.  424,  etc. 

9.  In  the  series  320,  318,  316,  etc.,  what  is  the  10th 
term?  21st?  31st?  41st?  Ans.  302,  etc. 

10.  In  the  series  412,  409,  406,  etc.,  what  is  the  15th 
term?  85th?  45th?  Ans.  370,  etc. 

11.  In  the  series  100,  95,  90,  etc.,  what  is  the  12th 
term?  15th?  20th?  21st?  Last  J[?is.  0. 

302.  To  find  the  common  difference^  when  the  ex- 
tremes and  the  numbers  of  terms  are  known. 

Divide  the  difference  of  the  extremes  by  the  number  of 
terms  less  1. 

EXAMPLES. 

1.  The  extremes  of  a  progression  are  4  and  151; 
number  of  terms  50.  What  is  the  common  diiference? 
What  is  the  series?  (151— 4)--(50— 1)=3,  the  common 
difference.     The  series  is  then  4,  7,  10,  etc. 


308  AlHTHMETICAL    PROGRESSION. 

2.  The  extremes  are  2  and  40 ;  number  of  terms  20. 
What  is  the  series?     Common  difference  is  2. 

Ans.  2,  4,  6,  8,  etc. 

3.  The  extremes  are  8  and  395 ;  number  of  terms  50. 
What  is  the  series?  Ans.  3,  11,  19,  etc. 

4.  Insert  8  means  between  the  extremes  7  and  43. 
(Vide  300,  (1.) 

Ans.  7,  11,  15,  19,  23,  27,  31,  35,  39,  43. 

5.  Insert  10  means  between  8  and  41. 

A71S.  8,  11,  14,  etc. 

6.  Insert  18  means  between  500  and  424, 

A71S.  500,  496,  etc. 

7.  Insert  8  means  between  320  and  302. 

8.  Insert  2  means  between  4  and  14. 

Ans.  4,  7J,  lOf,  14. 

9.  Insert  3  means  between  1  and  2. 

Ans.  1,  1J-,  11,  If,  2. 

10.  Insert  5  means  between  3  and  4. 

Ans.  3,  31,  3|,  etc. 

11.  Insert  7  means  between  3  and  6. 

Ans.  3,  3|,  3-|,  etc. 

12.  Insert  5  means  between  1  and  13. 

Ans.  1,  3,  5,  7,  9,  11,  13. 

14.  What  is  the  arithmetical  mean  between  4  and  10? 
(Vide  300,  (2.)  Ans.  7. 

15.  What  is  the  arithmetical  mean  between  1  and  2? 

Ans,  li=1.5. 

16.  What  is  the  arithmetical  mean  between  7  and  15? 

Ans.  11. 

17.  What  is  the  arithmetical  mean  between  1.5  and  1  ? 

Ans.  1.25 


ARITHMETICAL    PROGRESSION.  309 

303.  To  find  the  sum  of  all  the  terms,  when  the 
first  term,  the  last  term,  and  the  number  of  terms  are 
known. 

EXAMPLES. 

1.  The  extremes  are  1  and  13,  number  of  terms  7. 
What  is  the  sum  of  all  the  terms  ? 

ANALYSIS. 

We  have  just  found  the  series  to  be     (Ex.  12  above) 

1,      3,      5,      7,      9,    11,    13.  This  series 

reversed  is  13,    11,      9,      7,      5,      3,      1.    The  sum  of  double 

th«  series  is  then  1-1+144-14+14-f  14+14+14:=7Xl^=98,       and 

7V14 
the  sum  of  the  series  -itself  is  -^o —  =49,  as  may  be  verified  by 

adding  the  terms.     (Vide  300.)     ilence. 

Take  half  the  product  obtained  by  multiplying  the  sum 
of  the  extremes  by  the  number  of  terms. 

2.  What  is  the  sum  of  50  terms  of  the  series  4,  7, 
10,   etc.?      Last  term  is  151,  by  301. 

An,.  5^X^^=3875. 

3.  What  is  the  sum  of  20  terms  of  the  series  2,  4,  6, 
etc.?  40  terms?  60  terms?  71  terms?    Ans.  420,  etc. 

4.  What  is  the  sum  of  50  terms  of  the  series  3,  11, 
19,  etc.?  80  terms?  150  terms?  200  terms? 

^?2s.  9950,  etc. 

5.  What  is  the  sum* of  IQ  terms  of  the  series  7,  11, 
15,  etc.?  35  terms?  45  terms?   75  terms? 

6.  What  is  the  sum  of  12  t^rms  of  the  series  1,  2,  3, 4, 
etc.?  24  terms?  48  terms?  96  terms?     Ans.  78,  etc. 

7.  What  is  the  sum  of  2  terms  of  the  series  1,  3,  5, 
7,  etc.?  3  terms?  4  terms?  5  terms?  6  terms?  7  terms? 
25  terms?  Last  Ans.  625. 


310  ARITHMETICAL    niOGRESSION. 

8.  What  is  the  sum  of  40  terms  of  the  series  1,  4,  7, 
etc.?  50  terms?  60  terms?  70  terms? 

Ans.  2380,  etc. 

9.  A  gentleman  started  on  a  journey,  traveling  5 
miles  the  first  day,  7  miles  Jie  second  day,  and  so  on, 
gaining  2  miles  each  day.  Another  gentleman,  starting 
from  the  same  place,  travels  over  the  same  road  at  a 
uniform  rate  of  34  miles  per  day.  How  far  apart  were 
they  at  the  end  of  10  days?  20?  30? 

Last  Ans.  They  are  together. 

10.  A  gentleman  started  on  a  journey,  traveling  55 
miles  the  first  day,  51  the  second  day,  and  so  on,  losing 
4  miles  each  day.  Another  gentleman  travels  uniformly 
40  miles  per  day  over  the  same  road  and  from  the  same 
place.     How  far  apart  are  they  at  the  end  of  10  days? 

Ans.  30  miles. 

11.  A  boy  buys  12  marbles,  giving  1  cent  for  the  first, 
2  cts.  for  the  second,  3  cts.  for  the  third,  and  so  on. 
How  many  cents  did  he  give  for  all?         Ans.  78  cts. 

12.  Suppose  the  city  of  Quito  to  be  precisely  on  the 
equator,  and  that  this  circle  is  exactly  25000  miles  in 
circumference;  suppose,  furthermore,  that  the  earth 
were  entirely  of  land,  and  that  stakes  have  been  set  up 
at  intervals  of  1  mile,  tho  enti|;e  distance  round  the 
equator.  Now  allowing  a  bag  of  gold  dollars  to  be  at 
the  bottom  of  each  stake,  what  distance  would  a  person, 
setting  out  from  Quito,  h^v^e  to  travel  in  order  to  carry 
the  bags,  one  at  a  time,  to  that  city? 

A71S.  312500000  miles. 


GEOMETRICAL    PROGRESSION.  811 


GEOMETRICAL  PROGRESSION. 


304.  A  Geometrical  Progression  is  a  series  of  num- 
bers, in  which  any  term  is  found  by  multiplying  the  pre- 
ceding term  by  a  given  number. 

(1.)  The  ratio  is  the  number  used  as  a  multiplier. 

(2.)  The  progression  is  increasing  when  the  ratio  is 
greater  than  1. 

(3.)  The  progression  is  decreasing  when  the  ratio  is 
less  tha7i  1. 

305.  In  every  geometrical  progression  these  five 
points  may  be  considered,  viz.,  the  first  term,  the  last 
term,  the  ratio,  the  number  of  terms,  and  the  sum  of  all 
the  terms.     Thus,  in  the  series, 

1,  3,  9,  27,  81,  243, 
the  j^rs^  term  is  1,  last  term  ^4:^^^atio  3,  number  of  terms 
6,  sum  of  all  the  terms  364,  and  the  series  is  increasing. 
In  the  decreasing  progression, 

1     1     j_     _L    ^  ^  _    _  1  _ 

the  f.rst  term  is  1,  last  term  Y-0^24'  '^^^^^  h  'f^umber  of 
terms  6,  and  the  sum  of  all  the  terms  i§||. 

(1.)  The  first  and  last  terms  are  called  the  extremes, 
and  the  intermediate  terms  the  means. 

(2.)  In  a  series  of  three  terms  the  middle  term  is 
called  the  geometrical  mean,  and  is  the  square  root  of  the 
product  of  the  extremes. 

(3.)  The  ratio  is  the  quotient  of  any  term  divided  by 
the  preceding  term. 


312  GEOMETRICAL    PROGRESSION. 

306.  To  find  the  last  term,  when  the  first  term,  num- 
ber of  terms,  and  ratio  are  known. 

EXAMPLES. 

1.  If  the  fi7'st  term  of  a  series  is  2,  ratio  3,  and  num- 
her  of  terms  10,  what  is  the  last  term? 

ANALYSIS. 

If  we  take  a  few  terms  of  the  progression  thus, 
1st.  •    2d.  3d.  4th.  5tli. 

2,     2X3,     2X3^     2X3\     2X3^ 
that  is,       2         6  18  54  162 

we  see  that  any  term,  as,  for  instance,  the  5th,  is  found 
by  multiplying  the  first  term  by  that  power  of  the  ratio 
denoted  by  the  number  of  terms  less  1.  If  we  should 
continue  the  series  to  the  10th  term,  the  quantity  repre- 
senting ii  would  eviden{;|y  be  2X3^,  and  the  10th  term 
is  therefore  39366.     (Vj^je  ?76,  Ex.  1.)     Hence, 

Multiply  that  poiver  of  the  ratio  denoted  by  the  number 
of  terms  less  1  by  the  first  term. 

2.  In  the  series  3,  21,  147,  etc.,  what  is  the  4th  term? 
6th  term?  9th  term?     (Vid^e  273,  Ex.  13,  Table.) 

Last  Ans.  17294403. ' 

3.  In  the  series  4,  8,  16,  etc.,  what  is  the  8th  term? 
9th  term?  16th  terra?     (Vide  276,  Ex.  2.) 

Last  Ans.  131072. 

4.  In  the  series  2,  4,  8,  etc.,  what  is  the  5th  term  ? 
9th  terra?  13th  term?  Last  Aiis,  8192. 

5.  In  the  series  1,  J,  -},  etc.,  what  is  the  10th  term  ? 
6th  term?  4th  term?  Ans.  ^{.t,  etc. 


GEOMETRICAL    PROGRESSION.  313 

SOT.  To  find  the  ratio^  when  the  extremes  and  the 
number  of  terms  are  known, 

(1.)  Divide  the  last  term  by  the  first  term. 

(2.)  Take  the  root  of  the  quotient  denoted  by  the  number 
of  terms  less  1. 

EXAMPLES. 

1.  The  extremes  of  a  series  are  2  and  39366  ;  number 
of  terms,  10.  What  is  the  ratio  and  the  consequent 
series  ?  39366  --  2  =-- 19683.  Then  1^19683  =  3,  the 
ratio.     (Vide  273,  13,  Table ;  274,  (7.) 

•  The  series  is  2,  6,  18,  54,  162,  486,  1458,  4374, 
13122,  39366. 

2.  Insert  2  means  between  3  and  1029.  The  number 
of  terms  will  of  course  be  4.  1029 -r- 3 --=343;  then 
#"343=7,  the  ratio.  Ans.  3,  21,  147, 1029. 

3.  Insert  3  means  between  7  and  1792. 

Ans.  7,  28, 112,  448, 1792. 

4.  Insert  three  means  between  J  and  g^^- 

Ans.   i,  J^,  Jq,  Jq,  g*^. 

5.  What  is  the  geometrical  mean  between  4  and  16  ? 

Ans.  8.     (Vide  305,  (2.) 

6.  What  is  the  geometrical  mean  between  5  and  16 J? 

Ans.  9. 

7.  What  is  the  arithmetical  mean  between  1  and  0  ? 
(Vide  300,  (2.)  Ans.  .5 

8.  What  is  the  geometrical  mean  between  10  and  1  ? 

Ans.  3.162278. 

9.  What  is  the  arithmetical  mean  between  1  and  .5  ? 

Ans.  .75 

10.  What  is  the  geometrical  mean  between  10  and 
3.162278?  Ans.  5.623413. 

27 


314  GEOMETRICAL    PROaRESSION. 

11.  What  is  the  arithmetical  mean  between  .75  and  1  ? 

Ans.  .875. 
.  12.  What  is  the  geometrical  mean  between  10  and 
5.623413  ?  Ans.  7.498942. 

13.  What  is  the  arithmetical  mean  between  1   and 
.875?  Ans.  .9375. 

14.  What  is  the  geometrical  mean  between  10  and 
7.498942  ?  Ans.  8.659643. 

15.  Find  the  values  of  1±^  and  l/lOX 8.659643. 

16.  Find  the  values  of 

•''"'+•''''  and  1/9.305720x8.659643. 

17.  Find  the  values  of 

.96875+.953125  ^^^  i/9.305720x 8.976871. 

18.  Find  the  values  of 

■960988+.9a3i25  ^^^  ^9.139817x8.976871. 

19.  Find  the  values  of 

.957031+.953125  ^^^  ^-9.057978x8.976871. 

20.  Find  the  values  of 

.955078+.953125  ^^^  t/9.017333x8.976871. 

21.  Find  the  values  of 

.954102+.955078  ^^^  i/8.997079x9.017333. 

22.  Find  the  values  of 

.954590+.954102  ^^^  ^^9.007200X8.99707-9. 

28.  Find  the  values  of 


.954346+.954102 


---  and  1^9.002138X8.997079. 


and  1/9.000873x8.999608. 


GEOMETRICAL    PROGRESSION.  3^15 

24.  Find  the  values  of 

■?^^^^!^  and  •8.999608X.9002138. 

25.  Find  the  values  of 

.954285+.954224 
2 

26.  Find  the  values  of 

^'^^'Y'"^'  -d  1^9-000241X8.999608. 

27.  Find  the  values  of 
■954239+.904254  ^^^  ^8.999924x9.000241. 

28.  Find  the  values  of 

.9542474-.954235 


and  1^9.000082X8.999924. 
Ans.  .954243  and  9.000003. 

Remark. — Each  arithmetical  mean  in  the  above  examples  is  the 
Logarithm  of  the  corresponding  geometrical  mean,  beginning  with 
example  7.     The  logarithm  of  9  is  .954243.     (Vide  Algebra,  153.) 

308.  To  find  the  sum  of  all  the  terms,  when  the  ex- 
tremes and  ratio  are  given. 

EXAMPLES. 

1.  The  extremes  are  2  and  162,  the  ratio  3.  What  is 
the  sum  of  the  series? 

ANALYSIS. 

The  series  is  2,  2x3,  2X3^,  2x3^  2x3^  Multiplied 
by  3  it  is  2X3,  2X3^,  2x3^  2x3S  2X3^ 

The  first  series  added  would  give  the  sum  of  all  the 
terms ;  the  second  series  added  would  give  three  times 
the  sum  of  all  the  terms. 

If,  then,  the  first  series  be  subtracted  from  the  second, 


316  GEOMETRICAL    PROGRESSION. 

the  remainder  will  be  twice  the  sum  of  all  the  terms. 
The  actual  subtraction  gives  2X3^ — 2,  the  other  terms 
canceling  each  other.     Hence,  the  sum  of  the  series  is 

2 —  2"^^  -^^^-     Hence, 

Multiply  the  last  term  hy  the  ratio,  and  from  the  pro- 
duct subtract  the  first  term;  divide  the  remainder  by  the 
ratio,  less  1. 

2.  In  the  series  3,  21  .  .  .  1029,  what  is  the  sum  of 
all  the  terms?     (Yide  305,  (3.) 

Ans,  ^^^^f-^:^1200. 

3.  In  the  series  4,  12  .  .  .  78732,  what  is  the  sum  of 
all  the  terms?  Ans.  118096. 

4.  In  the  series  5,  20  .  .  .  327680,  what  is  the  sum 
of  all  the  terms?  Ans.  436905. 

5.  In  the  series  4,  8,  16,  etc.,  what  is  the  sum  of  16 
terms?     (Yide  306,  Ex.  3.)  J.ws.  262140. 

6.  In  the  series  2,  4,  8,  etc.,  what  is  the  sum  of  13 
terms?  Ans.  16382. 

7.  In  the  series  5,  50,  500,  etc.,  what  is  the  sum  of  8 
terms?  Ans.  55555555. 

309.  If  the  series  is  decreasing,  and  is  carried  on 
infinitely,  the  last  term  will  become  0.  Hence,  to  find 
the  sum  of  an  infinitely  decreasing  series. 

Divide  the  first  term  by  the  ratio  subtracted  from  1. 

EXAMPLES. 

1.  What  is  the  sum  of  the  series  J,  J,  ^,  etc.,  to  in- 
finity? Ans.  »^(1— i)=l. 

2.  What  is  the  sum  of  J,  y^^,  ^^5,  etc.?  Ans.  |. 

3.  What  is  the  sum  of  ^,  i,  ^,  etc.?  Am.  \. 


PERMUTATIONS.  817 

4.  What  is  the  sum  of  4,  1,  J,  y^^,  etc.?      Ans.  5-|. 

5.  What  is  the  value  of  .l-f-.Ol+.OOl,  etc.? 

Ans,  ^. 

6.  What  is  the  value  of  .1+.05+.025,  etc.? 


7. 
8. 

What 
What 

A 
is  the  value  of  l+f+g^,  etc.? 
is  the  value  of  |+/u+i3,  etc.? 

Vns.  i- 
Ans. 

Ans. 

=.2. 
If. 

2f. 

9. 

What 

is  the  value  of 

UA+t¥8,  etc. 

? 
Ans* 

If 

PERMUTATIONS 


310.  To  find  the  number  oi  permutations  that  can  be 
made  with  any  given  number  of  things,  each  one  differ- 
ent from  the  other, 

Multiply  all  the  consecutive  integral  numbers^  from  1 
up  to  the  given  number  of  things,  continually  together. 
The  product  will  be  the  permutations  that  can  be  made. 

EXAMPLES. 

1.  How  many  permutations  can  be  made  of  the  first 
four  letters  of  the  alphabet?     Ans.  1X2X3X4=24. 

2.  How  many  days  can  7  persons  be  placed  in  differ- 
ent positions  at  table?  Ans.  5040. 

3.  A  captain  of  26  men  told  his  company  that  he 
should  not  consider  them  perfectly  drilled  till  each  man 
had  occupied  all  possible  positions  in  the  arrangements 
that  might  be  made  of  them.     Suppose  his  men  drill  12 


318  PERMUTATIONS. 

hours  a  day,  and  make  a  change  every  hour,  how  many 
years  must  they  drill  before  becoming  perfect,  reckoning 
321  days  to  the  year? 

Ans.  107716786412020736000000  years. 

311.  To  find  how  many  arrangements  may  be  made 
by  taking  each  time  a  given  number  of  different  things 
less  than  all. 

Take  a  series  of  nwnhers  beginning  with  the  number 
of  things  given  and  decreasing  by  1  until  the  number  of 
terms  equals  the  number  of  things  to  be  taken  at  a  time; 
then  find  the  product  of  all  the  terms. 

EXAMPLES. 

1.  How  many  sets  of  3  letters  each  may  be  made  out 
of  4  letters  ?  Ans.  4  X  3  X  2==24. 

proof: 

abc,  acb,  adb,  bac,  bca,  bda,  cab,  cba,  cda,  dab,  dba,  dca, 

abd,  acd,  adc,  bad,  bed,  bdc,  cad,  cbd,  cdb,  dac,  dbc,  deb. 

2.  How  many  integral  numbers  can  be  expressed, 
each  composed  of  any  5  of  the  9  digits  ? 

Ans.  9X8X7X6X5=15120. 

3.  How  many  arrangements  can  be  made  out  of  the 
26  letters  of  the  alphabet,  6  being  taken  at  once? 

Ans.  165765600. 

312.  To  find  the  number  of  combinations  that  can  be 
made  of  any  number  of  things  in  sets  of  2  and  2;  3 
and  3,  etc., 

(1.)  Form  a  series  of  numbers^  as  in  311, /or  a  dividend. 

(2.)  Form  a  series  of  numbers^  as  in  310,  up  to  the 
number  of  things  to  be  combined  at  a  time,  for  a  divisor. 

The  quotient  will  be  the  number  of  combinations 
sought. 


PRACTICAL    GEOMETRY.  319 

EXAMPLES. 

1.  How  many  combinations  can  be  made  out  of  4 
letters,  having  3  different  letters  in  each  set? 

4X3X2 
^ns.   -^^2x3~'*- 
Proof. — abc,  abd,  acd,  bed. 

2.  How  many  combinations  can  be  made  out  of  10 
letters,  each  combination  having  in  it  7  letters,  but  no 
two  of  them  to  have  all  their  letters  alike  ? 

Ans.  120. 

3.  How  many  different  combinations  of  3  colors  can 
be  made  of  the  7  prismatic  colors?  Ans.  35. 

4.  How  many  different  combinations  of  4  colors  can 
be  made  of  the  7  prismatic  colors?  Ans.  35. 

5.  How  many  different  combinations  of  3  may  be 
made  of  10  different  things?  Ans.  120. 

6.  How  many  different  combinations  of  7  may  be 
made  of  10  different  things?  Ans.  120. 


PRACTICAL  GEOMETRY. 


DEFINITIONS. 

313.  Geometry  is  that  branch  of  Mathematics  which 
treats  of  the' relations  of  extension.  Extension  has  three 
dimensions — length,  breadth,  and  thichiess. 

(1.)  A  point  is  mere  position,  with  no  length,  breadth, 
or  thickness. 

(2.)  A  line  is  length,  without  breadth  or  thickness. 


320  PRACTICAL    GEOMETRY. 

(3.)  A  surface  is  a  figure  having  length  and  breadth, 
but  no  thickness. 

(4.)  A  solid  is  a  figure  having  length,  breadth,  and 
thickness. 

LINES. 

(5.)  A  straight  line  is  one,  all  points  of  which  lie  in 
the  same  direction.  It  is  designated  bj  letters  placed 
near  its  extreme  points.  Thus,  the  line  ab  is  repre- 
sented by  A B 

(6.)  A  curved  line  is  one,  ail  points  of  which  lie  in 
different  directions.     Thus,  the  curved  line 
A  B  is  represented  by 

(7.)  Parallel  straight  lines  are  two  or 

more   straight  lines   lying   in   the   same    c d 

direction.     Thus,  ab,  cd,  ef,  are  par-    ^  ^ 

allel  straight  lines. 

(8.)  Oblique  lines  are  those  which  do  not  lie  in  the 
same  direction.     Thus,  A  b   and  c  D  are    .    __ — _— — -^ 

oblique  lines.  c d 

Remark. — When  two  oblique  lines  meet  each  other,  they  form  an 
angle. 

(9.)  An  angle  is  the  divergence  of  two  b 

straight  lines  proceeding  from  the  same      ^^ o 

'point.     Thus,  a,  or  b  A  c,  or  c  A  b  is  aa 

angle.    In  reading,  the  letter  at  the  vertex     .        » 

is  placed  in  the  middle,  or  the  letter  at  / ^ 

the  vertex  may  be  used  alone  to  desig-           ^ 
nate  the  angle  when  other  angles  arc  not              ^^ 
adjacent.  ^^ o 


A B 


PRACTICAL    GEOMETRY.  321 

Eemark. — An  angle  is  greater  or  less^  according  as  the  lines  di- 
verge more  or  less. 

(10.)  A  right  angle  is  one  of  the  equal  angles  which 
two  straight  lines  may  make 
in   meeting   or   intersecting 

each  other.    Thus,  A  o  c,  B  o  c,  ^ 

A  0  D,  or  B  0  D  is  a  right  angle 
when  each  is  equal  to  any  other. 

An  acute  angle  is  one  which  is  less  than  a  right  angle. 

An  obtuse  angle  is  one  which  is  greater    •  „ 

than  a  rio;ht  angle.     Thus,  B  o  c  is  acute.  ^^ 

and  A  0  c  is  obtuse.  ° 

(11.)  Perpendicular  lines  are  those  which  form  a  right 
angle  with  each  other.     Thus,  A  o  is  per- 
pendicular  to  c  0,  and  c  o  is  perpendicular 
to  A  0,  when  A  o  c  is  a  right  angle. 


SURFACES. 

(12.)  A  plane  is  a  surface  in  which  if  any  two  points 
be  assumed  and  connected  by  a  straight  line,  that  line 
ivill  lie  wholly  in  the  surface.  Other  surfaces  are  called 
curved  surfaces.  Thus,  the  surface  of  a  common  slate 
represents  a  plane,  and  the  surface  of  a  slate  globe 
represents  a  curved  surface. 

(13.)  A  plane  figure  is  a  figure  bounded  by  straight 
or  curved  lines. 

(14.)  A  polygon  is  a  plane  bounded  by  straight  lines. 

(15.)  A  triangle  is  a  polygon  of  three  sides,  and  con- 
sequently of  three  angles. 

(a.)  An  equilateral  triangle  has  its  three  sides  equal; 


322 


PRACTICAL    GEOMETRY. 


an  isosceles  triangle  has  tivo  of  its  sides  equal;  a  scalene 
triangle  lias  its  three  sides  unequal, 

(b.)  A  right-angled  triangle  has  one  of  its  angles  a 
right  angle. 

(c.)  ArT  acute-angled  triangle  has  all  of  its  angles  acute. 

(e.)  An  obtuse-angled  triangle  has  one  of  its  angles 
an  obtuse  angle.     Thus : 


A  B  c  is  equilateral  and  acute-angled ;  D  E  F  is  isosceles 
and  right-angled;  mno  is  scalene  and  obtuse-angled, 
p  Q  R  is  scalene  and  acute-angled. 

(/.)  Any  side  of  a  triangle  may  be  assumed  as  the 
base,  and  its  altitude  is  the  perpendicular  line  extending 
from  the  vertex  of  the  opposite  angle  to  the  base. 

(g.)  In  a  right-angled  triangle  the  side  opposite  the 
right  angle  is  called  the  hypothenuse.  The  other  sides 
are  then  designated  as  the  base  and  perpendicular. 

(16.)  A  quadrilateral  is  a  polygon  of  four  sides. 

(17.)  A  parallelogram  is  a  quadrilateral  with  its  oppo- 
site sides  parallel. 

(18.)  A  rectangle  is  a  parallelogram 
having  a  right  angle. 

(19.)  A  rhomboid  is  a  parallelogram 
having  an  oblique  angle. 

(20.)  A  square  is  a  rectangle  hav- 
ing its  sides  all  equal. 

(21.)  A  rhombus  is  a  rhomboid  having 
its  sides  all  equal. 


S, ,Q 

a) |r 

Vi IB 


PRACTICAL    GEOMETRY.  323 

(22.)  A  trapezium  is  a  quadrilateral 
with  its  opposite  sides  7iot  parallel. 

(23.)  A   trapezoid  is  a  quadrilateral 
with  two  sides  parallel.     Thus: 

A  diagonal  is  a  line  joining  the  vertices  of  two  opposite 

angles.     Thus  T  E  is  a  diagonal. 

Remakk. — A  diagonal  divides  a  rectangle  into  two  right-angled 
triangles. 

(24.)  A  pentagon  is  a  polygon  of  five  sides;  a  hexagon, 
of  six  sides;  a  heptagon,  of  seven;  an  octagon,  of  eight; 
a  nonagon,  of  nine ;  and  a  decagon,  of  ten  sides. 

(25.)  A  circle  is  a  plane  figure 
bounded  by  a  curved  line  called  the 
circu7nference,  every  point  of  which 
is  equally  distant  from  a  point  within 
called  the  center.  The  diameter  is 
a  straight  line  drawn  through  the 
center  and  terminating  in  the  cir- 
cumference. A  radius  is  any  line  draivn  from  the  center 
to  the  circumference.  An  arc  is  a  part  of  the  circumfer- 
ence. Thus,  A  B  is  a  diameter,  o  B,  o  A,  and  o  M  are 
radii ;  A  M  is  an  arc. 

(26.)  Similar  figures  are  such  as  .are  mutually  equi- 
angular, and  have  the  sides  containing  the  equal  angles, 
taken  in  the  same  order,  proportional. 

314.  Proposition. — The  square  of  the  hypothenuse 
of  a  right-angled  triangle  is  equivalent  to  the  sum  of  the 
squares  on  the  other  two  sides.     Hence, 

315.  To  find  the  hypothenuse,  the  ttvo  sides  being  known, 
(1.)  Square  the  two  sides,  and  add  together  the  results. 
(2.)  Find  the  square  root  of  the  sum. 


324  PEACTICAL    GEOMETRY. 

EXAMPLES. 

1.  The  sides  of  a  right-angled  triangle  are  3  and  4 
feet.     What  is  the  length  of  the  hypothenuse  ? 

Ans.  -1/32  +  42  =  1/25  =  5  feet. 

2.  The  sides  of  a  right-angled  triangle  are  each  1 
foot.     What  is  the  length  of  the  hypothenuse  ? 

^Tis.  1/2  =  1.4142136  feet. 

3.  The  sides  of  a  right-angled  triangle  are  each  2  feet 
in  length ;  3  feet ;  4  feet ;  5  feet  in  length.  What  are 
the  corresponding  lengths  of  the  hypothenuse? 

Ans.  1^8  =  2.828427,  (Vide  286,  Ex.  1.) 

4.  The  bottom  of  a  window  is  40  feet  from  the  ground, 
and  I  wish  to  place  the  foot  of  a  ladder  30  feet  from  the 
bottom  of  the  wall  in  which  the  window  is  situated. 
What  is  the  length  of  a  ladder  that  will  reach  the  win- 
dow? Ans.  50  feet. 

5.  An  acre  of  land  is  laid  out  in  the  form  of  a  square. 
What  is  its  distance  between  opposite  corners  ? 

Ans.  1/320  =  8X1/5  =  17.888544  rods. 

6.  Suppose  a  floor  to  measure  16  feet  by  20.  What 
is  the  distance  between  opposite  corners  ? 

Ans.  1/656  =4X  l/41  =  25.612497. 

7.  What  is  the  length  of  the  lower  edge  of  a  brace 
which  touches  a  post  3^-  feet  from  the  corner,  and  a 
beam  4^  feet  from  the  same  point  ? 

Ans.  i  X  1/130  =  5.7008771  feet. 

316.  To  find  the  side  of  a  right-angled  triangle,  when 
the  hypothenuse  and  the  other  side  are  known. 

From  the  square  of  the  hypothenuse  subtract  the  square 
of  the  given  side.  The  square  root  of  the  remainder  will 
be  the  other  side. 


PRACTICAL   GEOMETRY.  325 

EXAMPLES. 

1.  The  hypothenuse  is  5  and  a  side  4.  Wliat  is  the 
other  side  of  the  triangle? 

2.  A  ladder  50  feet  long  is  placed  at  a  distance  of  30 
feet  from  the  bottom  of  a  house,  and  just  reaches  the 
sill  of  a  window.     What  is  the  hight  of  the  sill? 

Ans.  40  feet. 

3.  If  a  line  144  feet  long  will  reach  from  the  top  of  a 
fort  to  the  opposite  side  of  a  river  64  feet  wide,  on 
whose  brink  it  stands,  what  is  the  hight  of  the  fort  ? 

A71S.  129  feet,  nearly. 

4.  In  the  ruins  of  Persepolis  are  left  two  columns 
standing  upright ;  one  is  70  feet  high  and  the  other  50 
feet ;  in  a  line  between  them  stands  a  small  statue  5  feet 
high,  the  top  of  which  is  100  feet  from  the  summit  of 
the  higher,  and  80  feet  from  that  of  the  lower  column. 
What  is  the  distance  between  the  tops  of  the  two  col- 
umns ?  Ans.  143  J  feet. 

317.  To  find  the  side  of  a  square  when  its  diagonal 
is  known. 

Multiply  half  the  diagonal  by  the  square  root  of  2. 

EXAMPLES. 

1.  The  diagonal  of  a  square  is  1  foot.  What  is  the 
side  ?  Ans.  h  X  V2  =.7071068  feet. 

2.  The  diagonal  of  a  square  is  2  feet ;  3  feet ;  4  feet ; 
5  feet,  etc.     What  is  a  side?         Ans.  1.4142136,  etc. 

3.  The  distance  from  a  corner  to  an  opposite  corner 
of  a  square  room  is  20  feet.     What  is  a  side  ? 

Ans.  14.142136  feet. 


326  PRACTICAL    GEOMETRY. 

SIS.  Proposition. — In  any  triangle,  if  a  perpen- 
dicular be  drawn  from  its  vertex  to  its  base,  then  the 
whole  base  is  to  the  sum  of  the  other  two  sides  as  the  dif- 
ference of  those  sides  is  to  the  difference  of  the  parts  of 
the  base  made  by  the  perpendicular.  c 

Thus,  suppose  c  d  to  be  a  perpendicular 
drawn  from  the  vertex  c  to  the  base  A  B 
of  the  triangle  ABC,  then, 

A  B  :  A  c+B  c  : :  A  c— b  c  :  A  d— b  d. 

Remark. — The  perpendicular  must  always  fall  within  the  tri- 
angle. 

310.  To  find  the  perpendicular,  when  the  th'ee  sides 
of  a  triangle  are  known, 

(1.)  Make  the  base  the  first  terin  of  a  proportion;  the 
sum  of  the  other  tivo  sides  the  second  term;  the  difference 
of  the  two  sides  the  third  term.  The  fourth  term  found 
by  258  will  be  the  difference  of  the  parts  of  the  base. 

(2.)  Find  the  parts  by  125,  example  10,  (1)  and  (2). 

(3.)  Find  the  perpendicular  by  316. 

EXAMPLES. 

1.  In  a  triangle  ABC  we  have  ab=5  rods;  AC=4 
rods,  and  B  C— 3  rods.  What  is  the  length  of  the  per- 
pendicular CD? 

OPERATIONS. 

(1.)  5  :  4+3  : :  4-3  : 1.4. 

(2.)  (125,Ex.ll.)  -^±i:^=3.2-:ADand^::^=1.8=DB. 


(3.)  1/42-3.2^=2.4  or  1/32— 1.82=2.4=:D  c. 

Ans.  2.4  rods. 
Why  is  this  triangle  right-angled  at  c'i 


PRACTICAL   GEOMETET.  327 

2.  In  an  isosceles  triangle  the  sides  are  5,  5,  and  8 
rods.     What  is  the  length  of  the  perpendicular  ? 

Ans,  3  rods. 

Remark. — Assume  the  side  which  is  unequal  to  the  other  two  to 
be  the  base ;  then  the  parts  of  the  base  made  by  the  perpendicular 
will  be  each  4  rods. 

3.  In  an  equilateral  triangle  the  sides  are  each  1  rod. 
What  is  the  length  of  the  perpendicular  ? 

Ans.  |/r=i  =  Vi  =  hxVS  =  .8660254. 

4.  The  sides  of  a  triangle  are  4,  5,  and  6  rods.  What 
is  the  length  of  the  perpendicular? 

Ans.  3.307187  rods. 

5.  The  sides  of  a  triangle  are  10,  10,  and  16  rods. 
What  is  the  length  of  the  perpendicular?    Ans.  6  rods. 

6.  The  sides  of  a  triangle  are  each  25  rods.  What  is 
the  length  of  the  perpendicular?    Ans.  21.6506  rods. 

7.  The  sides  of  a  triano-le  are  each  2  rods  in  leno-th ; 
3  rods;  4  rods;  5  rods;  6  rods,  etc.  What  is  the 
length  of  the  perpendicular? 

Ans.l/S;  iXV^S;  2X1^3,  etc. 

MENSURA.TION. 

320.  Mensuration  is  the  method  of  finding  the 
area  of  surfaces  in  square  units  from  their  known  or 
implied  linear  dimensions.  By  mensuration  we  also  find 
the  cubical  contents  of  solids. 

321.  To  find  the  area  of  a  triangle,  when  its  sides 
are  known, 

(1.)  Find  the  perpendicular  by  319. 
(2.)  Multiply  the  base  by  the  perpendicular,  and  take 
half  the  product. 


328  PRACTICAL    GEOMETRY. 

EXAMPLES. 

1.  What  is  the  area  of  a  triangle  whose  sides  are  3, 

4,  and  5  rods?     (Vide  319,  Ex.  1.)     ?^,  or,  because 
the  triangle  is  right-angled,  -^ .  Ans.  6  rods. 

2.  What  is  the  area  of  a  triangle  whose  sides  are  5, 

5,  and  8  rods?  Ans.  12  rods. 

3.  What  is  the  area  of  a  triangle  whose  sides  are  each 
1  rod?  Ans.  .4330127  rods. 

4.  What  is  the  area  of  a  triangle  whose  sides  are  4, 
5,  and  6  rods?  Ans.  9.9215  rods. 

5.  What  is  the  area  of  a  triangle  whose  sides  are  10, 
10,  and  16  rods?  Ans.  48  rods. 

6.  What  is  the  area  of  a  triangle  whose  sides  are  each 
25  rods?  13  rods?  40  rods?     Ans.  270.633  rods,  etc. 

7.  How  many  square  feet  of  boards  will  cover  the 
gable  end  of  a  house  whose  rafters  measure  23  feet,  the 
length  of  the  beam  at  the  ends  of  the  building  being 
34.8712  feet?  Ans.  261.534  feet. 

Rule  II. — Add  together  the  three  sides,  and  taJce  half 
their  sum.  From  this  half  sum  subtract  each  side  sepa- 
rately; mtdtiply  together  the  half  sum  and  the  three  re- 
mainder.     The  square  root  of  the  product  will  be  the  area. 

8.  What  is  the  area  of  a  triangle  whose  sides  are  10, 
15,  and  20  rods?  15+10+20--45 ;  45-^2=22.5; 
V'22.5X2.5X12.5X7.5==72.62.         Ans.  72.62  rods. 

333.  To  find  the  area  of  any  quadrilateral,  two  of 
whose  sides  are  parallel, 

Multiply  the  sum  of  the  parallel  sides  hy  the  perpendic- 
ular distance  between  them,  and  take  half  their  product. 


PRACTICAL    GEOMETRY.  329 

EXAMPLES. 

1.  How  many  yards  of  carpeting  will  cover  a  floor  20 
feet  long  and  16  feet  Avide,  carpet  1  yard  wide?  20 X 
16-^-9.  Ans.  35 f  yards. 

2.  How  many  acres  in  a  rectangular  piece  of  land 
17  chains  long  and  5  ch.  and  41.  wide?  (Vide  180, 
Rem.  2.)     17x5.04^10         Ans.  8  A.  2  R.  10/o'(j  r- 

3.  How  many  acres  in  a  piece  of  land  in  the  form  of 
a  rhombus,  each  side  measuring  70  rods  and  the  width 
being  8  rods  ?  Ans,  3  A.  2  R. 

4.  How  many  acres  in  a  square  piece  of  land  whose- 
sides  are  each  35  ch.  25  1.     '        Ans.  124  A.  1  R.  1  r. 

5.  A  board  measures  25  feet  in  length  and  1  ft.  6  in. 
in  width.     What  is  the  area  in  feet?      Ans.  37 J  feet. 

6.  In  the  trapezoid,  313  (23,)  I  D  is  20  rods;  z  o, 
27  rods;  P  R,  12  rods.  How  many  acres  in  the  lot? 
m^}><}l  Ans.  1  A.  3  R.  2  r. 

7.  In  a  trapezoid  the  parallel  sides  are  25  ch.  13  1. 
and  30  ch.  1 1. ;  the  perpendicular  distance  between  them 
is  40  ch.     How  many  acres  ?     Ans.  110  A.  1  R.  4.8  r. 

S23.  To  find  the  area  of  a  trapezium,  when  all  its 
sides  and  a  diagonal  are  known, 

Find  the  area  of  each  triangle  hy  321,  and  their  sum 
will  be  the  area  of  the  trapezium. 

EXAMPLES. 

1.  In  a  trapezium  represented  by  the  figure  ZIUM, 
813,  (22,)  z I  is  25  rods;  mi,  23  rods;  zu,  17  rods; 
U  M,  21  rods ;  and  z  m,  30  rods.     How  many  acres  ? 

Ans.  2A.  3R.  13.7  r. 

28 


330  PRACTICAL    GEOMETRY. 

324.  Any  iivo  similar  figures  have  their  areas  in  pro- 
portion to  the  squares  of  any  ttvo  lines  similarly  situated 
in  each.  Hence,  all  circles  are  to  each  other  as  the  squares 
of  their  radii,  or  the  squares  of  their  diameters. 

325.  To  find  the  area  of  a  circle, 

Multiply  half  of  the  diameter  by  half  the  circumfer- 
ence. 

Remark. — The  circumference  of  a  circle  whose  diameter  is  1  is 
3.141592053589793238462643383279502884197169399375105820974- 
94459230781640628620899862803482534211706798214808651327230- 
66470938446,  etc. 

EXAMPLES. 

1.  What  is  the  area  of  a  circle  whose  diameter  is  1  ? 

Ans.  .7853981633974483,  etc. 
Hence,  to  find  the  area  of  a  circle,  midtiply  the  square 
of  the  diameter  hy  enough  of  the  figures  composing  the 
decimal  .785398163397,  etc.,  to  make  the  area  sufficiently 
exact. 

2.  What  is  the  area  of  a  circle  whose  diameter  is  2 
rods?  3  rods?  4  rods?  5. rods?  etc. 

Ans.  3.1416,  etc.,  rods;  7.06858,  etc.,  rods. 

3.  What  is  the  area  of  a  circle  whose  diameter  is  20 
rods?  30  rods?  40  rods?  etc. 

4.  What  is  the  area  of  a  circle  whose  radius  is  10 
rods?  15  rods?  20  rods?  etc. 

5.  What  is  the  area  of  a  circle  whose  radius  is  100 
rods?  150  rods?  etc. 

6.  What  is  the  area  of  a  circle  whose  radius  is  5000 
rods?  Ans.  490873  A.  3  R.  16  r. 

7.  What  is  the  area  of  a  circle  whose  radius  is  1 
mile?  2  miles?  Smiles?  etc.    J.?is.  3.1416  miles,  etc. 


rRACTICAL    GEOMETRY.  331 

8.  What  is  the  area  of  a  circle  whose  radius  is  5 
yards?  6  yards?  etc.  Ans.  78 J  yards,  etc. 

9.  Three  men  purchase  a  grind- 
stone with  a  radius  of  30  inches.        '' /  \''~''"-^/\\ 
How  much  of  the  radius  must  each      ,' 
grind  oiF  to  secure  J  of  the  stone,      \ 
making  no   allowance  for  loss   at       ''  ^ 
the  center?     We  have  ""-" 


Circ.  A  0  : 

:  circ.  c  o  : 

:  AO^ 

:  C  0^ ;  that  is, 

1   ; 

:        1         : 

:  30-  ; 

:30^Xf. 

Circ.  A  0  ; 

:  circ.  P  0  : 

:  AO^ 

:  P  0- :  that  is, 

1 

-•        i        -• 

:  30-^  : 

;  30^X^ 

Since  c  0^=30^X1,  c  o=30Xl/|-=10Xl/6--24.4949. 
Since  p  o2=30-Xl,  p  o=30Xi/1=10XV^3=17.3205. 

Hence,  the  fii?st  takes  30—24.4949=5.5051  inches; 
the  second  takes  24.4949—17.3205=7.1744  inches;  the 
third  takes  17.3205  inches. 

Remark. — It  is  only  necessary  to  use  the  ratio  of  the  .areas  in 
the  proportions. 

10.  Five  men  purchased  a  grindstone,  and  each  man 
paid  ^  of  the  price.  The  radius  was  40  inches.  How 
much  of  it  must  each  man  grind  off  to  secure  his 
share?  Ans,  First  man  40— 16x/5=4.2229  in. 

11.  I  have  a  circular  garden  25  rods  in  diameter, 
and  wish  to  make  a  walk  around  it  that  shall  take  up 
I  of  the  entire  area.  What  must  be  the  width  of  the 
walk?  Ans.  1.0891  rods. 

12.  I  have  a  circular  garden  50  rods  in  diameter,  and 
wish  to  make  a  walk  around  the  outside  of  it  whose  area 
shall  be  J^  the  area  of  the  garden.  What  must  be  the 
width  of  the  walk?  Ans.  1.22  rods. 


332 


PRACTICAL    GEOMETRY. 


MENSURATION    OF    SOLIDS. 

326.  Demnition. — (1.)  A  cylinder  \^  a  solid  described 
by  the  revolution  of  a  rectangle  about  one  of  its  sides, 
which  remains  fixed. 

(2.)  A  cone  is  a  solid  described  by  the  revolution  of 
a  right-angled  triangle  about  one  of  its  sides^  which  re- 
mains fixed. 

(3.)  A  sphere  is  a  solid  described  by  the  revolution 
of  a  semicircle  about  its  diameter,  which  remains  fixed. 
Thus : 


0  P  B  D  revolved  about  o  P  generates  a  cylinder ;  c  0  B 
revolved  about  c  o  generates  a  cone ;  and  A  M  B  revolved 
about  A  B  generates  a  sphere,  o  p  is  the  altitude  of  the 
cylinder,  o  c  of  the  cone,  and  A  B  is  the  diameter  of  the 
sphere. 

(4.)  A  prism  is  a  solid  whose  bases  are  j[?<2raZ?eZ  and 
its  sides  parallelograms.  A  right  prism  has  its  edges 
perpendicular  to  its  bases, 

(5.)  A  parallelopiped  is  a  prism  whose  bases  as  well 
as  sides  are  parallelograms.  A  cube  has  six  equal  square 
faces. 

(6.)  A  pyramid  is  a  solid,  having  a  polygon  for  a 
base  and  three  or  more  triangles  for  its  sides,  whose 


PRACTICAL    GEOMETllY. 


333 


vertices  meet  in  a  common  point  called  the  vertex  of 
the  pyramid.  The  frustrum  of  a  pyramid  is  the  part 
left  after  cutting  off  a  portion  of  the  top  by  a  plane 
parallel  to  the  base.     Thus  : 


/^.   \ 


\ 

\o 

\ 

F 

D 
1 

li 

A  B  c-D  E  F  is  a  right  triangular  prism ;  A  B  c  D-E  is  a 
right  quadrangular  prism,  or  parallelopiped;  A  B  c-D  is 
a  triangular  pyramid ;  and  A  B  c-s  represents  the  frus- 
trum of  a  pyramid. 

3S7.    To  find  the  contents  of  a  cylinder  or  prism, 
Multiply  the  area  of  the  base  by  its  altitude. 

EXAMPLES. 

1.  Each  side  of  the  base  of  a  triangular  prism  is  1 
foot,  and  the  altitude  is  3  ft.  2  in.     What  are  the  con- 
tents in  feet?  in  inches?     (Vide  321,  Ex.  3.) 
.4330127X31.  Ans.  1.3712  feet. 

2.  The  sides  of  a  triangular  prism  are  7,  8,  and  9 
inches.  Its  altitude  is  Ij  feet.  Find  the  contents  in 
inches.  Ajis.  482.99  inches. 

3.  The  diameter  of  each  end  of  a  cylinder  is  8  feet, 
and  the  hight  is  5  J  feet.     Find  the  contents  in  feet. 

Alts.  276.46  feet. 


334  PRACTICAL    GEOMETRl. 

4.  The  diameter  of  a  cylindrical  water-pail  is  10 
inches,  and  the  hight  is  1  foot.  How  many  wine  gal- 
lons does  the  pail  hold?     (Vide  182,  Rem.  1.) 

A71S.  4.08  gallons. 

5.  How  many  bushels  in  a  box  15  feet  long,  5  feet 
wide,  and  8  feet  deep?     (Vide  184,  Rem.) 

Ans.  482.142  bushels. 

6.  How  many  bushels  in  a  box  6  feet  long,  1^  feet 
wide,  and  2 J  feet  deep?  Ans.  IS. OS. 

7.  What  are  the  contents  in  cubic  feet  of  a  wall  24 
feet  3  inches  long,  10  feet  9  inches  high,  and  2  feet 
thick?  Ans.  521|. 

3S8.  To  find  the  contents  of  a  cone  or  pyramid, 
Multiply  the  area  of  the  base  by  the  altitude^  and  take 
I  of  the  product. 

EXAMPLES. 

1.  Each  side  of  the  base  of  a  triangular  pyramid  is  1 
foot,  and  the  hight  is  14  inches.  What  are  the  con- 
tents? Ans.  290.9844  in. 

2.  The  sides  of  the  base  of  a  triangular  pyramid  are 
10,  11,  and  12  feet,  and  the  hight  is  12  feet.  What  are 
the  contents?  Ans.  206.085  feet. 

3.  The  base  of  a  cone  is  10  feet  in  diameter  and  the 
hight  is  5  feet.     Find  the  contents. 

Ans.  130.899  feet. 

4.  A  square  pyramid,  477  feet  high,  has  each  side  of 
its  base  720  feet  in  length.     Find  the  contents. 

Ans.  3052800  cu.  yd. 

5.  The  sides  of  the  base  of  a  triangular  pyramid, 
which  is  14J  feet  high,  are  5,  6,  and  7  feet.  Find  the 
contents.  Ans.  71.0352  feet. 


PRACTICAL    GEOMETRY.  335 

3S9.  To  find  the  contents  of  the  frustrum  of  a  cone 
or  pyramid, 

Find  the  sum  of  the  areas  of  the  two  ends  and  the  geo- 
metrical mean  between  them.  Multiply  this  sum  hy  the 
altitude,  and  take  one  third  of  the  product 

EXAMPLES. 

1.  A  stick  of  timber  is  15  in.  square  at  one  end  and 
6  in.  square  at  the  other  and  24  feet  long.  What  arQ 
the  contents?      (^^5+B«+«0)X8  ^„,.- 19,  feet. 

2.  A  conic  frustrum  is  18  feet  high,  8  feet  in  diam- 
eter at  one  end  and  4  feet  at  the  other.  Find  the 
contents.  Ans.  527.7888  feet. 

3.  A  cask,  in  the  form  of  two  equal  conic  frustrums, 
has  a  bung  diameter  of  28  inches,  a  head  diameter  of  20 
inches,  and  a  length  of  40  inches.  How  many  gallons 
of  wine  will  it  hold?  Ans.  79.0613  gallons. 

7.  A  cistern  is  12  feet  in  diameter  at  the  top,  10  feet 
in  diameter  at  the  bottom,  and  14  feet  deep.  What 
number  of  gallons  will  it  hold? 

OPERATION. 
Area  of  top        =     144  X     -7854  =  113.09 
Area  of  bottom  .-=     100  X     .7854=     78.54 
Geomet.  mean    =t/113.09X78.54  =     94.25 

Sum  =  285.88  feet. 
Then  285.88XV*X1728--231     =9980      gallons,  nearly,  ^hs. 

8.  How  many  Winchester  bushels  will  the  above  cis- 
tern hold?  A71S.  1072  bushels. 

330.  Proposition. — All  spheres  are  to  each  other  as 
the  cubes  of  their  radii,  or  diameters. 


336  MISCELLANEOUS    EXAMPLES. 

331.  To  find  the  surface  and  also  the  solid  contents 
of  a  sphere, 

(1.)  Multiply  the  square  of  the  diameter  by  3.1415^, 
eic.^  for  the  surface. 

(2.)  ^lultiply  the  cube  of  the  diameter  hy  .523598,  etc.^ 
for  the  solid  contents. 

Remark. — The  latter  decimal  is  i  of  the  former. 

EXAMPLES. 

1.  An  artificial  globe  is  24  inches  in  diameter;  what 

is  its  area,  and  what  the  solid  contents  ? 

Arts.  Area  1809.556992  sq.  in.;  solidity  2738.218752  cu.  in. 

2.  A  slate  globe  is  6  feet  in  diameter;  what  is  its 
area,  and  what  its  contents  ? 

Ans.  Area  113.0973  sq.ft.;  contents  113.0973 cu.  ft. 

3.  A  sphere  is  40  feet  in  diameter;  what  will  be  the 
diameter  of  one  containing  |  as  many  cubic  feet?  J-  as 
many?  i  as  many?     (Vide  296,  Ex.  3,  2,  and  1.) 

Ans.  40X^^1=36.34  ft.;  40X'^  1=31.75  ft.;  40X#'i=25.19  ft. 


332.  MISCELLANEOUS  EXAMPLES. 


1.  If  a  certain  number  be  multiplied  by  5,  and  the 
product  divided  by  J,  and  3  be  added  to  the  quotient, 
and  7  taken  from  the  sum,  the  remainder  will  be  76. 
What  is  the  number  ?  Ans.  8. 

2.  If  to  a  certain  number  12  be  added,  and  the  square 
root  of  the  sum  taken,  the  cube  of  that  root  will  be  64. 
What  is  the  number  ?  Ans,  4. 


^ 


APPENDIX. 


353 


III. 

TABLE  OP  CUBE  ROOTS. 

No. 

1 

Cube  Root. 

No. 

Cube  Root. 

No. 

Cube  Root. 

No. 

Cube  Root. 

1.000000 

Ig" 

3.583048 

91 

4.497941 

136 

5.142563 

2 

1.259921 

47 

3.608826 

92 

4.514357 

137 

6.155137 

3 

1.442250 

48 

3.634241 

93 

4.530655 

138 

5.167649 

4 

1.587401 

49 

3.659306 

94 

4.546836 

139 

5.180102 

5 

1.709976 

50 

3.684031 

95 

4.562903 

140 

5.192494 

6 

1.817121 

51 

3.708430 

96 

4.5J8857 

141 

5.204828 

7 

1.912931 

52 

3.732511 

97 

4.594701 

142 

5.217103 

8 

2.000000- 

53 

3.756286 

98 

4.610436 

143 

5.229322 

9 

2.080084 

54 

3.779763 

99 

4.626065 

144 

5.241483 

10 

2.154435 

55 

3.802953 

100 

4.641589 

145 

5.253588 

11 

2.223980 

56 

3.825862 

101 

4.657009 

146 

5.265637 

12 

2.289429 

57 

3.848501 

102 

4.672329 

147 

5.277632 

13 

2.351335 

58 

3.870877 

103 

4.687548 

148 

5.289573 

14 

2.410142 

59 

3.892996 

104 

4.702669 

149 

5.301459 

15 

2.466212 

60 

3.914868 

105 

4.717694 

150 

5.313293 

16 

2.519842 

61 

3.936497 

106 

4.732624 

151 

5.325074 

17 

2.571282 

62 

3.957892 

107 

4.747459 

152 

5.336803 

18 

2.620742 

63 

3.979057 

108 

4.762203 

153 

5.348481 

19 

2.668402 

64 

4.000000 

109 

4.776856 

154 

5.360108 

■  20 

2.714418 

65 

4.020726 

110 

4.791420 

155 

5.371685 

21 

2.758924 

66 

4.041240 

111 

4.805896 

156 

5.383213 

22 

2.802039 

67 

4.061548 

112 

4.820285 

157 

5.394691 

23 

2.843867 

68 

4.081655 

113 

4.834588 

158 

5.406120 

24 

2.884499 

69 

4.101566 

114 

4.848808 

159 

6.417502 

25 

2.924018 

70 

4.121285 

115 

4.862944 

160 

6.428835 

2G 

2.962496 

71 

4.140818 

116 

4.876999 

161 

5.440122 

27 

3.000000 

72 

4.160168 

117 

4.890973 

162 

5.451362 

28 

3.036589 

73 

4.179339 

118 

4.904868 

163 

5.462556 

•29 

3.072317 

74 

4.198336 

119 

4.918685 

164 

5.473704 

30 

3.107233 

75 

4.217163 

120 

4.932424 

165 

5.484807 

31 

3.141381 

76 

4.235824 

121 

4.946087 

166 

5.495865 

32 

3.174802 

77 

4.254321 

122 

4.959676 

167 

5.506878 

33 

3.207534 

78 

4.272659 

123 

4.973190 

168 

5.517848 

34 

3.239612 

79 

4.290840 

124 

4.986631 

169 

5.528775 

35 

3.271066 

80 

4.30^70 

125 

5.000000 

170 

6.539658 

36 

3.301927 

81 

4.326749 

126 

5.013298 

171 

5.550499 

37 

3.332222 

82 

4.344482 

127 

5.026526 

172 

5.561298 

38 

3.361975 

83 

4.362071 

128 

5.039684 

173 

5.672055 

39 

3.391211 

84 

4.379519 

129 

5.052774 

174 

5.582770 

40 

3.419952 

85 

4.396830 

130 

5.065797 

175 

5.593445 

41 

3.448217 

86 

4.414005 

131 

5.078753 

176 

6.604079 

42 

3.476027 

87 

4.431048 

132 

5.091643 

177 

6.614672 

43 

3.503398 

88 

4.447960 

133 

5.104469 

178 

6.625226 

44 

3.530348 

89 

4.464745 

134 

5.117230 

179 

6.635741 

i  45 

3.556893 

90 

4.481405 

135 

5.129928 

180 

5.646216 

354 


APPENDIX. 


IV. 


TABLE: 


Shoiving  the  ultimate  fratisveise  strength  of  a  bar  1  foci  long  and  1  inch 

1  inch  in  diameter,  made  of  either  of  the  materials  mentioned.     The  bar  is  loaded 
in  the  middle,  and  lies  loose  at  both  ends. 


Matebials. 

Square  Bar 

Onk  Third. 

Round  Bar. 

One  Third. 

Oak 

800 

1137 

o69 

916 

600 

2580 

4013 

269 
379 
189 
305 
200 
860 
1338 

628 
893 
447 
719 
471 
2026 
3152 

209 
298 
149 
239 
157 
675 
1050 

Ash 

Elm 

Pitch-pine 

Pine 

Wroui'lit-iron 

To  find  the  ultimate  transverse  strength  of  any  rectangular 
beam,  supported  at  both  ends  and  loaded  in  the  middle, 

Multiply  the  strength  of  an  inch  square  bar  1  foot  long,  as  in  the 
Table,  by  the  breadth,  and  by  the  square  of  the  depth,  in  inches,  and 
divide  the  product  by  the  length,  in  feet. 

The  quotient  will  be  the  weight  in  avoirdupois  pounds. 

Remark  1. — When  a  beam  is  supported  in  the  middle  and  loaded  at  each  end, 
it  will  bear  the  same  weiglit  as  when  supported  at  both  ends  and  loaded  in  the 
middle  ;  that  is,  each  end  will  boar  half  the  weight. 

Remark  2. — When  the  Aveight  is  applied  somewliero  between  the  middle  and 
the  end  of  the  beam,  multiply  twice  the  length  of  the  long  end  by  twice  the  length  of 
the  short  end,  and  divide  the  product  by  the  whole  length  of  the  beam.  The  quotient^, 
is  the  effective  length  of  the  beam. 

Remark  3. — If  the  beam  is  round,  multiply  the  ultimate  strength  of  the  round  bar, 
in  the  Table,  by  the  cube  of  the  diameter,  in  inches,  and  divide  the  product  by  the 
length,  IN  FKET. 

Remark  4. — When  a  beam  is  fixed  at  both  ends  and  loaded  in  the  middle,  it 
will  bear  one  half  more  than  when  loose  at  both  ends.  If  the  beam  is  loose  at 
both  ends,  and  tho  weight  is  applied  uniformfy  along  its  length,  it  will  bear 
double;  but  if  fixed  at  both  ends,  and  tho  weight  applied  uniformly  along  its 
length,  it  will  bear  triple  tho  weight. 


EXAMPLES. 


1.  What  weight  will  break  a  beam  of  asli  5  inches  broad,  7  inches 
deep,  and  26  feet  deep  between  the  supports? 

1137X^X7^ 


Ans. 


:111431b.,  nearly. 


APrENDix.  355 

2.  What  is  the  ultimate  strength  of  an  oak  Lcam  20  feet  long, 
4  inches  broad,  8  inches  deep,  and  the  weight  placed  G  feet  from 

the  end?      ??^  =10.8  feet.     (Vide  Rem.  2.)     Then, 

800X4X8^^^,,,^,,^     ^,,. 
lb,8 

3.  What  is  the  ultimate  transverse  strength  of  a  wrought-iron 
solid  cylinder,  10  feet  long  and  5  inches  in  diameter  ? 

Ans.         "-^    =39400  lb.  (Vide  Rem.  3.) 


ANNUITIES. 

An  Annuity  is  an  estate  which  entitles  its  owner  to  the  pay- 
ment of  a  fixed  sum,  at  regular  intervals  of  time. 

The  annuity,  time,  and  rate  of  interest  being  given,  to  find  the 
amount, 

Raise  the  ratio  to  a  power  denoted  by  the  time,  from  which  subtract  1; 
divide  the  remainder  by  the  ratio  less  1,  and  multiply  the  quotient  by  the 
annuity.     The  product  will  be  the  amount. 
Remabk  1. — For  the  powers  of  the  ratio,  see  Table  under  249. 

EXAMPLES. 

1.  What  is  the  amount  of  a  pension  of  $100  per  annum,  which 
has  remained  unpaid  for  5  years,  interest  6  per  cent.? 

1.06^=1.338226.     Then  .338226--.06Xl00=$563.71.  Ans. 

2.  What  is  the  amount  of  an  annual  rent  of  $150,  in  arrears 
for  12  years,  at  6  per  cent,  compound  interest? 

Ans.  $2530.489999. 

3.  Wliat  is  the  amount  of  a  pension  of  $900,  in  arrears  17  years, 
at  7  per  cent,  compound  interest?  Ans.  27756.193. 

4.  What  is  the  amount  of  an  annual  salary  of  $6000,  in  arrears 
for  8  years,  at  5  per  cent,  compound  interest? 

Ans.  $57294.60. 

5.  What  is  the  amount  of  a  $100  pension,  in  arrears  20  years, 
at  5  per  cent.?  6  per  cent.?  7  per  cent.? 

Ans.  $3306.596 ;  $3678.558 ;  $4099.55. 
G.  What  is  the  amount  of  a  pension  of  $1000,  in  arrears  for  12 
years,  at  7  per  cent,  compound  interest?  Ans.  $17888.45. 


356 


APPENDIX. 


The  annuity,  time,  and  rate  of  interest  being  given,  to  find  the 
present  worth, 

Divide  the  amount^  as  found  ahove^  hy  the  ratio  raised  to  a  power 
denoted  hy  the  time. 

EXAMPLES. 

1.  What  is  the  present  worth  of  a  pension  of  $100,  to  continue 
5  years,  at  6  per  cent,  per  annum? 

^563.71-f-1.0G'^=$421.24.  Ans. 
Remauk  2. — In  this  way  the  following  Table  may  be  constructed : 


v.  — TABLE: 

Shoioing  the  presetit  value  of  $1.00  for  any  number  of  years,  from  1  to  25,  at  5,  G, 
7,  8,  and  10  per  cent. 


Yeaes 

5  Per  Cent. 

6  Per  Cent. 

7  Per  Cent. 

8  Per  Cent. 

10  Per  Cent. 

1 

0.952381 

0.943396 

0.934579 

0.925926 

0.909091 

2 

1.859410 

1.833393 

1.808018 

1.783265 

1.735537 

3 

2.723248 

2.673012 

2.624316 

2.577097 

2.486852 

4 

3.545951 

3.465106 

3.387211 

3.312127 

3.169865 

5 

4.329477 

4.212364 

4.100197 

3.992710 

3.790787 

6 

5.075692 

4.917324 

4.766540 

4.622880 

4.455261 

7 

5.786373 

5.582381 

5.389289 

5.206370 

4.868419 

8 

6.463213 

6.209794 

5.971299 

5.746639 

5.334926 

9 

7.107822 

6.801692 

6.515232 

6.246888 

5.759024 

10 

7.721735 

7.360087 

7.023582 

6.710081 

6.144557 

11 

8.306414 

7.886875 

7.498674 

7.138964 

6.495061 

12 

8.863252 

8.383844 

7.942686 

7.536078 

6.813692 

13 

9.393573 

8.852683 

8.357651 

7.903776 

7.103356 

14 

9.898641 

9.294984 

8.745468 

8.244237 

7.366687 

15 

10.379658 

9.712249 

9.107914 

8.559479 

7.606080 

16 

10.837770 

10.105895 

9.446649 

8.851369 

7.823701 

17 

11.274066 

10.477260 

9.763223 

9.121638 

8.021553 

18 

11.689587 

10.827603 

10.059087 

9.371887 

8.201412 

19 

12.085321 

11.158116 

10.335595 

9.603599 

8.364920 

20 

12.462210 

11.469921 

10.594014 

9.818147 

8.513564 

21 

12.821153 

11.764077 

10.835527 

10.016803 

8.648694 

22 

13.W3003 

12.041582 

11.061241 

10.200744 

8.771540 

23 

13.488574 

12.303379 

11.272187 

10.371059 

8.883218 

24 

13.798642 

12.550358 

11.469334 

10.528758 

8.984744 

25 

14.093945 

12.783356 

11.653583 

10.074776 

9.077040 

2.  What  is  the  present  worth  of  a  pension  of  $800,  to  continue 
25  years,  at  10  per  cent.?  Ans.  $2723.11. 


APPENDIX. 


357 


MISCELLANEOUS  TABLE. 

12  units make 1  dozen. 

12  dozen "  1  gross. 

12  gross "  1  great  gross. 

20  units "  1  score. 

100  years "  1  century. 

10  centuries "  1  chiliad. 

100  pounds "  1  quintal  of  fisli. 

196  pounds "  1  barrel  of  flour. 

200  pounds "  1  barrel  of  pork. 

14  pounds "  1  stone, 

21J  stones "  1  pig- 

8  pigs "  1  fother. 

18  inches "  1  cubit. 

6  feet "  1  fathom. 

24  sheets "  1  quire. 

20  quires r...  "  • 1  ream. 

2  reams "  1  bundle. 

5  bundles "  1  bale. 


FRENCH  WEIGHT. 

Frexch  weight  is  that  used  in  the  empire  of  France.  Its  units 
are  named  milligramme,  centigramme,  decigramme,  gramme,  decagramme, 
hectogrmme,  kilogramme,  myriagramme,  quintal,  millier  or  bar.  The 
milligramme  is  the  unit  of  lowest  value,  and  the  bar  the  highest. 
It  takes  10  of  any  order  to  make  one  of  the  next  higher  order  of 
units,  except  that  100  quintals  make  1  bar. 

The  GRAMME  is  the  fundamental  unit,  and  is  the  weight  of  a 
centimetre  of  pure  water,  at  the  temperature  of  melting  ice,  which 
is  15.43402  grains  Troy. 

1  quintal  =  1  cwt.  3  gr.  25  lb.;  1  millier  or  bar  =  9  T.  16  cwt. 
3  gr.  12  lb. 


1   pound   Avoirdupois  =  453| 


1   pound  Troy 
grammes. 

FRENCH  LINEAR  MEASURE. 

The  units  of  this  measure  are  named  millimetre,  centimetre,  deci- 
metre, METRES,  decametre,  hectometre,  kilometre,  and  myriametre;  and 
it  takes  10  units  of  any  lower  order  to  make  1  of  the  next  higher. 


358  APPENDIX. 

The  METRE  is  the  fundamental  unit,  and  one  of  the  ten  million 
equal  parts  into  which  the  meridian  distance  from  the  equator 
to  the  north  pole  is  divided.  This  distance  is  39.37079  English 
inches.  It  is  thence  easy  to  find  the  value  of  any  French  unit  in 
English  measure. 

FRENCH  SUPERFICIAL  MEASURE. 

The  units  of  this  measure  are  named  milliare,  centiare^  deciare, 
ARE,  decare,  hectare,  kilare,  and  myrlare;  and  it  takes  10  units  of  a 
lower  order  to  make  1  of  the  next  higher. 

The  ARE  is  the  fundamental  unit,  and  is  a  square  decametre^ 
which  is  equivalent  to  1076.4298  square  feet. 

FRENCH  SOLID  MEASURE. 

The  units  of  this  measure  are  named  centistere,  decistere,  stere, 
and  decastere;  and  it  takes  10  units  of  a  lower  order  to  make  1  of 
the  next  higher. 

The  stere  is  the  fundamental  unit,  and  is  a  cubic  metre,  which 
is  equivalent  to  35.3174  cubic  feet. 

FRENCH  MEASURE  OF  CAPACITY. 

The  units  of  this  measure  are  named  millitre,  centilitre,  decilitre, 
LITRE,  decalitre,  hectolitre,  kilolitre,  and  myrialitre. 

The  LITRE  is  the  fundamental  unit,  and  is  a  cubic  decimetre, 
which  is  61.027051  cubic  inches. 

1  litre  =  1|  English  pints;   1  hectolitre  =  22  English  gallons. 

1  decalitre  =  2.6414308  wine  gallons;  1  hectolitre =2.834  Win- 
chester bushels. 

RATES  OF  FOREIGN  MONEY  OR  CURRENCY. 

(FIXED    BY    LAW.) 

Ducat  of  Naples $0  80 

Florin  of  the  Netherlands 40 

Florin  of  the  Southern  States  of  Germany 40 

Florin  of  Austria  and  Trieste 48J 

Florin  of  Nuremburg  and  Frankfort 40 

Florin  of  Bohemia 48^ 

Guilder  of  the  Netherlands 40 

Lira  of  Lombardo  and  the  Vcuetiuu  Kingdom 16 


ArpENDix.  3-59 

Livre  of  Leghorn $0  16 

Lira  of  Tuscany 16 

Lira  of  Sardinia 18f 

Livre  of  Geneva 18f 

Milrea  of  Portugal 1  12 

Milrea  of  ]\Ladeira 1  00 

Milrea  of  Azores 83i- 

Marc  Banco  of  Hamburg 35 

Ounce  of  Sicily 2  40 

Pound  Sterling  of  Jamaica 4  84 

Pound  Sterling  of  the  British  Provinces 4  00 

Pagoda  of  India 1  84 

Real  Vellon  of  Spain 05 

Real  Plate  of  Spain 10 

Rupee  of  British  India 44| 

Rix  Dollar  (or  Thaler)  of  Prussia  and  North  Germany..       69 

Thaler  (or  Rix  Dollar)  of  Bremen 78| 

Thaler  (or  Rix  Dollar;  of  Berlin,  Saxony,  and  Leipsic...       69 

Rouble  (Silver)  of  Russia 75 

Specie  Dollar  of  Denmark 1  05 

Specie  Dollar  of  Norway 1'06 

Specie  Dollar  of  Svreden 1  06 

Tale  of  China 1  48 

Banco  Rix  Dollar  of  Sweden  and  Norway 39| 

Banco  Rix  Dollar  of  Denmark 53 

Crown  of  Tuscany 1  05 

Curacoa  Guilder 40 

Leghorn  Dollar  or  Pezzo ^Oj^iny 

Livre  of  Catalonia 53| 

Livre  of  N\ifchatel 26| 

Swiss  Livre 27 

Scudi  of  Malta 40 

Roman  Scudi 99 

St.  Gall  Guilder 403-^^^ 

Rix  Dollar  of  Batavia 75 

Roman  Dollar 1  05 

Turkish  Piastre 05 

Current  Mark 28 

Florin  of  Prussia..! , 22f 


360  APPENDIX. 

Florin  of  Basle $0  41 

Genoa  Livre 21 

Livre  Tournois  of  France. 18| 

Rouble  (paper)  of  Russia,  varies  from  4j^^^  to  i^^^jj  to  the  dollar, 

BOOKS  AND  PAPER. 

NAMES  AND  SIZES  OF  PAPER  MADE  BY  MACHINERY. 

Double  Imperial 32    by  44  inches. 

Double  Superroyal 27    by  42  " 

Double  Medium 23    by  26  " 

"  "       24    by  37i  « 

"  "       25    by  38  " 

Royal  and  Half 25    by  29  " 

Imperial  and  Half 26    by  32  « 

Imperial 22    by  32  « 

Superroyal 21    by  27  « 

Royal 19    by  24  « 

Medium ^. ^ 181  by  23|  « 

Demy ." 17    by  22  " 

Folio  Post ,. ♦ 16    by  21  " 

Fo?4^5ap.....^...':..:?^^ ,| 14    by  17  « 


fo-wn 


Cro-wl....c:r:../ ^!V.... 15    by  20       " 

^  sheet  folded  in  2  leaves  is  called  -dT  folio. 

A  sheet  ''  4  '  ""             "     ^^  quarto  or  4to, 

A  sheet  "  8  ''■"             "an  octavo  or  8vo. 

'           A'jshee'f  ,    "  12  "  "^      "         a  12rao. 

A  sheet  "  18  "             "         an  18mo. 

A  sheet  "  24  "            **         a  24mo. 

A  sheet  «  32  »             "         a  32mo. 


THE    END. 


s 


m  17439 


M305987 

T  5'9 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


m  ■ 

liliiiililiill' 


■n 


-3^ 


I 


J 


